# Thread: Colin Leslie Dean Crackpottery

1. GiantEvil said
Mr edam421, if Mr. Kurt Godel's incompleteness theorem is wrong, pick an axiom and prove it.
colin leslie dean points out godels theorem is invalid-illegitimate because it uses the invalid axiom of reducibility

http://www.scribd.com/doc/32970323/G...d-illegitimate

regardless of how faultless godels proof/logic is
his proof is invalid as it uses the invalid axiom of reducibility

godel axiom 1v is the axiom of reducibility
and his formula 40 uses the axiom of reducibility

IV. Every formula derived from the schema
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free. This axiom represents the axiom of reducibility (the axiom of comprehension of set theory)
“ [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u & u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution “(K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965)
ramsy wittgenstien russel etc say this axiom is invalid

godel tells us that in his proof of the incompletness theorem he uses axiom 1v
NOTE HE SAYS PROOF

In the proof of Proposition VI the only properties of the system P employed were the following:
1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of Proposition V).
Hence in every formal system that satisfies assumptions 1 and 2 and is ω-consistent, undecidable propositions exist of the form (x) F(x), where F is a recursively defined property of natural numbers, and so too in every extension of such
system p is

“P is essentially the system obtained by superimposing on the Peano axioms the logic of PM”

AXIOMS OF P
“I.
Gödel uses only three of the Peano postulates; the others are supplanted by the axion-schemata defined later.
1. ~(Sx1 = 0)
Zero is the successor of no number. Expanded into the basic signs, the axiom is: ~(a2 ∀ (~(a2(x1)) ∨ a2(0)))
This is the smallest axiom in the entire system (although there are smaller theorems, such as 0=0).
2. Sx1 = Sy1 ⊃ x1 = y1
If x+1 = y+1 then x=y. Expanding the ⊃ operator we get: ~(Sx1 = Sy1) ∨ (x1 = y1) And expanding the = operators we get: ~(a2 ∀ (~(a2(Sx1)) ∨ a2(Sy1))) ∨ (a2 ∀ (~(a2(x1)) ∨ a2(y1)))
3. x2(0).x1 ∀ (x2(x1) ⊃ x2(fx1)) ⊃ x1 ∀ (x2(x1))
The principle of mathematical induction: If something is true for x=0, and if you can show that whenever it is true for y it is also true for y+1, then it is true for all whole numbers x.
[178]II. Every formula derived from the following schemata by substitution of any formulae for p, q and r.
1. p ∨ p ⊃ p
2. p ⊃ p ∨ q
3. p ∨ q ⊃ q ∨ p
4. (p ⊃ q) ⊃ (r ∨ p ⊃ r ∨ q)
III. Every formula derived from the two schemata
1. v ∀ (a) ∨ Subst a(v|c)
2. v ∀ (b ⊃ a) ∨ b ⊃ v ∀ (a)
by making the following substitutions for a, v, b, c (and carrying out in I the operation denoted by "Subst"): for a any given formula, for v any variable, for b any formula in which v does not appear free, for c a sign of the same type as v, provided that c contains no variable which is bound in a at a place where v is free.23
IV. Every formula derived from the schema
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free. This axiom represents the axiom of reducibility (the axiom of comprehension of set theory).
V. Every formula derived from the following by type-lift (and this formula itself):
1. x1 ∀ (x2(x1) ≡ y2(x1)) ∨ x2 = y2.
This axiom states that a class is completely determined by its elements.”

http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.

"The system P of footnote 48a is Godel’s
streamlined version of Russell’s theory of types built on the natural
numbers as individuals, the system used in [1931]. The last sentence ofthe footnote allstomindtheotherreferencetosettheoryinthatpaper;
KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing
his approach to set theory, “This axiom plays the role of [Russell’s]
axiom of reducibility (the comprehension axiom of set theory).”

NOW RAMSEY RUSSELL WITGENSTIEN say the axiom of reducibility is invalid
“As a corollary, the axiom of reducibility was banished as irrelevant to mathematics ... The axiom has been regarded as re-instating the semantic paradoxes” - http://mind.oxfordjournals.org/cgi/r...07/428/823.pdf
4) Russell Ramsey and Wittgenstein regarded it as illegitimate Russell abandoned this axiom – in 2nd ed PM- and many believe it is illegitimate and must be not used in mathematics

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

the standford encyclopdeia of philosophy says of AR

http://plato.stanford.edu/entries/pr...a-mathematica/

“many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically”

From Kurt Godels collected works vol 3 p.119

“the axiom of reducibility is generally regarded as the grossest philosophical expediency “
also godel used the second edition of PM
but in that edition russell repudiated the axiom- so godels proof uses a version of PM that russell abandoned

“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), and the axioms of reducibility
IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY IN THAT EDITION – which Godel must have known. Godel used a text in PM that based on Russells revised version of PM in 2nd ed PM Russell had rejected abandoned dropped as stated in the introduction. Godel used a text with the axiom of reducibility in it but Russell had abandoned rejected dropped this axiom as stated in the introduction. Godel used a rejected text as it used the rejected axiom of reducibility.

The Cambridge History of Philosophy, 1870-1945- page 154

Quote

“In the Introduction to the second edition of Principia, Russell repudiated Reducibility as 'clearly not the sort of axiom with which we can rest content'…Russells own system with out reducibility was rendered incapable of achieving its own purpose”

quote page 14
http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

“Russell gave up the Axiom of Reducibility in the second edition of
Principia (1925)”

Phenomenology and Logic: The Boston College Lectures on Mathematical Logic and Existentialism (Collected Works of Bernard Lonergan) page 43
“In the second edition Whitehead and Russell took the step of using the simplified theory of types dropping the axiom of reducibility and not worrying to much about the semantical difficulties”
In Godels collected works vol 11 page 133

it says AR is dropped
quote

In the second edition of Principia (at least in the introduction) ...the axiom of reducibility is
dropped

2.

3. Well done edam, now a few questions.

When an odd man speak:
1. should tax money go to someone that speak so complex that we break?

You are welcome to prove anything wrong for me, but I wouldn't give a dang if I can't read it.

4. put even more simply

regardless of how faultless godels proof/logic is

his proof is invalid as it uses the invalid axiom of reducibility

godel axiom 1v is the axiom of reducibility
and his formula 40 uses the axiom of reducibility
and this axiom is invalid-according to ramsey wittgenstien russell and most others

For instance this one :

No proposition can say anything about itself, because the proposition sign cannot be contained in itself (that is the "whole theory of types").

6. No proposition can say anything about itself, because the proposition sign cannot be contained in itself (that is the "whole theory of types").

that statement is about impredicative statements
Poincare Russell Wittgenstien and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics

dean points out that godel uses impredicative statement in his proof thus making his proof again invalid

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

[quote]Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves[/quote]r

thus making his proof invalid

yes wittgenstein said AR was invalid

http://en.wikipedia.org/wiki/Axiom_of_reducibility
This problem appears later when Wittgenstein arrives at this gentle disavowal of the axiom of reducibility --
[/url]

7. My problem is English is not my native language and this is not the easiest to read.
So to give you right or wrong is not possible and also not contributing. There still seems to be argument about as it shows....

What helps me in this matter is to understand and use a subtle difference between showing (something shows (visual) or makes hear, smell etc.
and proving. Somethings shows often without proof being necessary (so no right or wrong). For instance with music. "this music is great, beautifull, exiting etc." the musical experience is what it is to someone, no need to proof it, the music does not have to proof itself nor does the musician has to proof his music other then by playing. As the music has impact on lives and thus live it is real and no proof (in words) necessary. So there is no truth involved that way (you can,t speak about it Wittgenstein would say) yet it can be true i.e. true music as it makes hear but it does not deliver a proof in words as it is a different language (allthough spoken language has nuances and tones similar to music). It is true as it influences live(s)to a deeper level. A painting also just shows.

Same way I make this distinction between proof and showing with pythagoras. What is referred to as the evidence for the formula to me is not evidence. Other way round neither ; the geometry does not proof the formula. It just translates it in the context of an isolated/particular system where all lines between the corners are drawn in ink. If you look at it in that geometrical context (drawing) and study. A^2+b^2=C^2 it is true but if the diagonal is not drawn and the perpendiculars are forces ; How do you know if it has the same logic outcome there may be other similar forces of influence when it is no longer in isolation.

We can wonder about it in a way that is not that much different from art or music, it shows to an extend.

Prooving often is about a translation to another realm of communication ; music explained in to words for example or even spoken language translated to written. Geometry translated to algebra etc. So the question maybe should be if that doesn,t limit by definition.

I a way A^2+b^2=C^2 is a system also ; particular. For instance you have to know that the squares reflect surfaces. Otherwise it has no meaning at all and a lot of people can tell you that A^2+b^2=C^2 (it is a truth for them) but it has no meaning for them at all.

Same way 2*3=6 will be regarded as true by most people. And if you ask them how much is 1*6 ? they would answer its the same.

So you have six piles and lay them in one row of six and another 6 piles you lay in two rows of three. Then ask is this the same as that ? They will answer no thats different. Then you ask so is 1*6 the same as 2*3 ? Huh ? eh yes eh no eh don,t know.
It becomes clear if you understand that
In 2+3 both numbers can stand for apple but in 2*3 not. Because what is a scquared apple ? So either the 2 or the 3 stands for apples and the other number for an activity like three apples in a row twice or putting two apples in a row thrice.

So if 2*3=6 would be true and 1*6=6 also then because 6 ecquals 6, 1*6 would ecqual 2*3.

6 = 6 is also not a truth. One can be apples and the other can be pears but also one can be six granny's and the other six golden delicious. Ore one six golden delicious and the other six different golden delicious.
6 and 6 are two different blobs of ink also. So the discussion is extremely difficult.

And maybe making statements like Russel and/or Wittgenstein prooved this or that is not wise in this. Just as making a statement for Godells theorems as being truths. You have to wonder if it is in the spirit of these people that you take there statements as truths. The statements are made in a context to the person and discussions at that time also. So maybe they can,t be prooven or disprooven to be true.
If you argue against Godells theorems you go from the idea that for Godell these theorems where supposed to be considered as thruths. You can also put up a theorem for discussion.
Often for the followers a theorem or something becomes a truth but that says little. You would not be arguing with Godell then but with his followers.

8. I might be wrong about this as my contacts with algebra have been fairly shallow and are a few years old, but haven't Wittgenstein's ideas on Godel's theorem been disapproved? I remember there was a big fuss a few years ago, re-stirred by Floyd and Putnam but their work has been quite heavily criticized by logicians and has had relatively little impact.

Analyzing these statements takes a lot of care because, as far as axiomatic systems go, it's fairly difficult to get to a universal truth. Wittgenstein's thoughts were carefully considered by the logicists of his time because of his very influential Tractatus. Contemporary peer-review was, however, not very bright on his statements over the incompleteness theorem and Godel himself was dissatisfied with what he considered to be an incorrect analysis of his work from Wittgenstein's part, as the latter did not correctly understand some of the theorem's more intricate parts.

Granted, I think you're far better informed on these topics than I am, but you might want to do a bit of research on arguments that disprove Wittgenstein's conclusions, so that you may get a clearer picture of the field.

9. Granted, I think you're far better informed on these topics than I am, but you might want to do a bit of research on arguments that disprove Wittgenstein's conclusions, so that you may get a clearer picture of the field.
many people have said axiom of reducibility is invalid
4) Russell Ramsey and Wittgenstein regarded it as illegitimate Russell abandoned this axiom – in 2nd ed PM- and many believe it is illegitimate and must be not used in mathematics

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

the standford encyclopdeia of philosophy says of AR

http://plato.stanford.edu/entries/pr...a-mathematica/

“many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically”

From Kurt Godels collected works vol 3 p.119

“the axiom of reducibility is generally regarded as the grossest philosophical expediency “
as noted russell himself droped repudiated abandoned axiom of reducibility from the 2ed of PM -the very version godel said he used

IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY IN THAT EDITION – which Godel must have known. Godel used a text in PM that based on Russells revised version of PM in 2nd ed PM Russell had rejected abandoned dropped as stated in the introduction. Godel used a text with the axiom of reducibility in it but Russell had abandoned rejected dropped this axiom as stated in the introduction. Godel used a rejected text as it used the rejected axiom of reducibility.

The Cambridge History of Philosophy, 1870-1945- page 154

Quote

“In the Introduction to the second edition of Principia, Russell repudiated Reducibility as 'clearly not the sort of axiom with which we can rest content'…Russells own system with out reducibility was rendered incapable of achieving its own purpose”

quote page 14
http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

“Russell gave up the Axiom of Reducibility in the second edition of
Principia (1925)”

Phenomenology and Logic: The Boston College Lectures on Mathematical Logic and Existentialism (Collected Works of Bernard Lonergan) page 43
“In the second edition Whitehead and Russell took the step of using the simplified theory of types dropping the axiom of reducibility and not worrying to much about the semantical difficulties”
In Godels collected works vol 11 page 133

it says AR is dropped
quote

In the second edition of Principia (at least in the introduction) ...the axiom of reducibility is
dropped

10. colin leslie dean points out that godel uses impredicative statement in his proof thus making his proof again invalid

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

Poincare Russell Wittgenstien and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics

Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves

11. Originally Posted by edam421
colin leslie dean points out that godel uses impredicative statement in his proof thus making his proof again invalid

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

Poincare Russell Wittgenstien and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics

Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves
Colin Leslie Dean is an idiot, so there is no point in quoting him.

The only axioms used by Godel are the axioms that are required to enable ordinary arithmetic of the natural numbers.

There are no restrictions on definitions. You are describing a philosophical debate, involving, not the use of some additional axiom, like the "axiom of reducibility" but rather the imposition of such an additional axiom. Mathematics is based largely on the Zermelo-Fraenkel axioms plus the axiom of choice. In that system there is no such thing as the "axiom of reducibility".

http://en.wikipedia.org/wiki/Zermelo...kel_set_theory

http://www.research.ibm.com/people/h...n00-goedel.pdf

What we have here is a true dichotomy. Kurt Godel was a genuine genuis. Colin Leslie Dean is a genuine idiot.

12. Originally Posted by DrRocket
Originally Posted by edam421
colin leslie dean points out that godel uses impredicative statement in his proof thus making his proof again invalid

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

Poincare Russell Wittgenstien and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics

Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves
Colin Leslie Dean is an idiot, so there is no point in quoting him.

The only axioms used by Godel are the axioms that are required to enable ordinary arithmetic of the natural numbers.

There are no restrictions on definitions. You are describing a philosophical debate, involving, not the use of some additional axiom, like the "axiom of reducibility" but rather the imposition of such an additional axiom. Mathematics is based largely on the Zermelo-Fraenkel axioms plus the axiom of choice. In that system there is no such thing as the "axiom of reducibility".

http://en.wikipedia.org/wiki/Zermelo...kel_set_theory

http://www.research.ibm.com/people/h...n00-goedel.pdf

What we have here is a true dichotomy. Kurt Godel was a genuine genuis. Colin Leslie Dean is a genuine idiot.
Hey rocket in ass, don't they teach you adequate measures?

13. Originally Posted by LeavingQuietly
Hey rocket in ass, don't they teach you adequate measures?
Did you forget your medication this morning ?

14. There are no restrictions on definitions. You are describing a philosophical debate, involving, not the use of some additional axiom, like the "axiom of reducibility"
in godels proof the axiom of reducibility is not some additional axiom
but is integral to HIS proof
it is is axiom 1v of his system P
and formula 40
and because AR is an invalid axiom
his theorem is invalid

15. Originally Posted by edam421
There are no restrictions on definitions. You are describing a philosophical debate, involving, not the use of some additional axiom, like the "axiom of reducibility"
in godels proof the axiom of reducibility is not some additional axiom
but is integral to HIS proof
it is is axiom 1v of his system P
and formula 40
and because AR is an invalid axiom
his theorem is invalid
Ok he does have a version of the axiom of replacement or what is also called the comprehension axiom. All it says is that if you have a set and a class of elements taken from that set then the class is also a set.

If you have trouble with that then you have trouble with all of mathematics and all of logic

Since you have shown that you have trouble with mathematics and no ability in logic, I am not surprised that you have trouble with this. End of story.

Godels theorems are quite valid.

Colin Leslie Dean is clearly an idiot. You seem to want to follow in his footsteps.

16. Ok he does have a version of the axiom of replacement or what is also called the comprehension axiom. All it says is that if you have a set and a class of elements taken from that set then the class is also a set.

If you have trouble with that then you have trouble with all of mathematics and all of logic
you say Ok

and because he uses the invalid axiom of reducibility- as ramsey wittgenstien and even russell and others admit
his theorem is invalid

even the editors of godels collected works critice the axiom
From Kurt Godels collected works vol 3 p.119

“the axiom of reducibility is generally regarded as the grossest philosophical expediency

17. Colin Leslie Dean is an idiot
you keep calling dean an idiot in many different threads

and that has nothing to do with his arguments but your psychological make up
here is the reason

like most people you try and put the pieces of your reality together into an ordered whole
where there are no cracks and you cant see the joins
but what dean does
is shows you your ordered world is really fall of big holes ie it ends in meaninglessness
now that really throws you into some sort of anxiety- your ordered world now falls apart
thats why you dont like dean

18. If you bother to read the whole paragraph you quoted, you will see that it is not the editors who have a problem with Godel's work, but Russell.

edam421, I think what you don't get here is the role of axioms per se.

What CLD is contesting is the validity of the axiom of reducibility from a philosophical point of view. This is utterly irrelevant to the conclusion of Godel's theorem, because it does not apply to everything in this world but only to those systems where this axiom is assumed to be correct. Which happens to include pretty much every piece of mathematics. Regardless of whether Russell agrees with the axiom or not, Godel's conclusions apply to every piece of mathematics ever conceived which holds the axiom of reducibility to be true.

One is free to debate the validity of every axiom, of course; for instance, one can think of a space where Euclid's axiom is not true. However, this does not make Euclidean geometry incorrect in those spaces where the axiom is assumed to be true.

Also, please remember that axioms are assumed, not proven to be true, in a given system.

Bottom line: in every mathematical system where the axiom of separation is assumed to be true, Godel's conclusion stands. Wittgenstein's and Russell's systems do not assume that the axiom of separation is correct, and in those frameworks, Godel's theorem does not hold. It does hold in any other system that assumes it to be true, which (as far as I remember) includes ZFC, upon which set theory, number theory, vector space theory and a lot of other branches of mathematics are built.

19. What CLD is contesting is the validity of the axiom of reducibility from a philosophical point of view. This is utterly irrelevant to the conclusion of Godel's theorem, because it does not apply to everything in this world but only to those systems where this axiom is assumed to be correct. Which happens to include pretty much every piece of mathematics. Regardless of whether Russell agrees with the axiom or not, Godel's conclusions apply to every piece of mathematics ever conceived which holds the axiom of reducibility to be true.
many others have said AR is invalid ie ramsey etc

Ramsey says

Such an axiom has no place in mathematics
it is about it being invalid as it uses an invalid axiom

you can assume as long as you like that the axiom is correct
fact is
it is invalid -thus his theorem is invalid

you can assume 1+1=3
and come up with an incomprehension theorem
big deal
all you will be showing is how cleaver you are
big deal

with no bearing or relevance to a real world where 1+1=2

Godel's conclusions apply to every piece of mathematics ever conceived which holds the axiom of reducibility to be true.
wrong
godels conclusion is meaningless

ie there are true mathematical statements which cant be proven

as godel cant tell us what makes a maths statement true,
thus his theorem is meaningless nonsense

just like if from my theorem based on 1+1=3
i said there are gibble statements which cant ge proven
if i cant tell u what a gibble statement is
then my theorem would be laughed at and rejected for being meaningless
just as godels theorem is
as colin leslie dean has shown
go see the thread on this point
[/quote]

20. Originally Posted by AlexandruLazar

Bottom line: in every mathematical system where the axiom of separation is assumed to be true, Godel's conclusion stands. Wittgenstein's and Russell's systems do not assume that the axiom of separation is correct, and in those frameworks, Godel's theorem does not hold. It does hold in any other system that assumes it to be true, which (as far as I remember) includes ZFC, upon which set theory, number theory, vector space theory and a lot of other branches of mathematics are built.
Yes.

In fact, outside of logic, essentially all of mathematics is built on Zermelo Fraenkel. The axiom of choice is also nearly universal, outside of Errett Bishop's constrctivist school, and I don't think there is much work in that area any longer. There is zero controversy on this. In most mathematical discourse ZFC is accepted as the basis without any comment whatever.

There is no such thing as an "invalid axiom" so long as the system of axioms is not provably inconsistent. Axioms are, by definition, statements that are taken as true without proof. The problem with inconsitent axioms is that anything, literally any statement that can be formulated, is both true and fale in an inconsistent system.

It seems to me to be goiing a bit far to say that Russel rejected the axiom of reducibility (what you are calling the axiom of separation). He invented it. Later he criticized it, but so far as I know never rejected it.

The reason for the axiom, in simple terms, is simply to avoid Russel's paradox, which arises from the consideration of classes that are "too big" to be sets -- like the "set of all sets" -- which result in the paradox.

But you are absolutely correct Godel's theorem is valid and applies to any system that meets the criteria described in Godel's paper -- and that is ALL of mathematics as it is commonly understood.

21. I also do not know of Russell rejecting the axiom of separation (I also know it under this name actually) -- but given my limited familiarity with this field I assumed egan421 is right on Russell's attitude since he covered more material on the subject than I have (I am familiar with ZFC but I have limited knowledge of the entire historical context in which it was developed). I remember reading that Russell later considered the axiom of separation not to be the best solution, but that's about all I know about how it was criticized.

22. It seems to me to be goiing a bit far to say that Russel rejected the axiom of reducibility (what you are calling the axiom of separation). He invented it. Later he criticized it, but so far as I know never rejected it.
Russell repudiated Reducibility

The Cambridge History of Philosophy, 1870-1945- page 154

Quote

“In the Introduction to the second edition of Principia, Russell repudiated Reducibility as 'clearly not the sort of axiom with which we can rest content'…Russells own system with out reducibility was rendered incapable of achieving its own purpose”

quote page 14
http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

“Russell gave up the Axiom of Reducibility in the second edition of
Principia (1925)”

Phenomenology and Logic: The Boston College Lectures on Mathematical Logic and Existentialism (Collected Works of Bernard Lonergan) page 43

“In the second edition Whitehead and Russell took the step of using the simplified theory of types dropping the axiom of reducibility and not worrying to much about the semantical difficulties”
In Godels collected works vol 11 page 133

it says AR is dropped
quote

In the second edition of Principia (at least in the introduction) ...the axiom of reducibility is dropped

There is zero controversy on this. In most mathematical discourse ZFC is accepted as the basis without any comment whatever.
the axiom of choice was/is controversial
ZFC uses the axiom of seperation which is impredicative and such statements are is controversal

to keep your world ordered you seem to be in denial about controversial things

There is no such thing as an "invalid axiom" so long as the system of axioms is not provably inconsistent
most people argue AR is invalid
as shown russell repudiated it
ramsey outlawed it from mathematics
others have said similar things

as i said i can make 1+1=3
to be true by definition
and construct a theory about gibbles
big deal
it would be laughed out of the country
same with godel and his invalid theorem about truth -which he cant tell us what that is

23. AFAIK, Russell dropped the axiom of separation from his second edition not because he considered it to be inconsistent, but because he did not consider the semantic problems which result from not including it to be too important. On the other hand, it introduced a number of additional constraints which Russell could not solve in his system, but which are solved in ZFC.

as i said i can make 1+1=3
to be true by definition
and construct a theory about gibbles
No, you cannot make 1+1=3 by definition because that would break the axiom of extensionality and you would work in an inconsistent environment.

24. AFAIK, Russell dropped the axiom of separation from his second edition not because he considered it to be inconsistent, but because he did not consider the semantic problems which result from not including it to be too important. On the other hand, it introduced a number of additional constraints which Russell could not solve in his system, but which are solved in ZFC.
so what
as has been said others even ramsey say it is invalid-most say it is invalid
in this world AR is invalid
thus in this world godels theorem is invalid

but which are solved in ZFC
i would not use that system as a bench mark
an ad hoc system useing an ad hoc impredicative axiom ie axiom of seperation
using an axiom of choice which was/is controversial
s

25. i would not use that system as a bench mark
You may not use that system as a benchmark, but every piece of mathematics is built on it.

26. You may not use that system as a benchmark, but every piece of mathematics except for logic is built on it.
so what
it uses axioms which are controversal ie axiom of choice
the axiom of seperation is just an ad hoc tag on to stop the russell paradox
it is impredicative thus making it s leititamacy in question
over all even if logic is built on it-it is built on unsteady ground

and further many argue impredicative statements should be outlawed
but a lot of maths is built on such statements
thus making your whole maths look ridiculous
further
maths assumes it is consistent
but when paradoxes arise showing it is inconsistent
mathematician just invent an ad hoc add on ie axiom of seperation to save the day

are you coping DrRocket= no heart failure yet

27. so what
Well, nothing, except that since most fields of mathematics are based on it, Godel's theorem applies to them.

28. Originally Posted by edam421
You may not use that system as a benchmark, but every piece of mathematics except for logic is built on it.
so what
it uses axioms which are controversal ie axiom of choice
the axiom of seperation is just an ad hoc tag on to stop the russell paradox
it is impredicative thus making it s leititamacy in question
over all even if logic is built on it-it is built on unsteady ground

and further many argue impredicative statements should be outlawed
but a lot of maths is built on such statements
thus making your whole maths look ridiculous
further
maths assumes it is consistent
but when paradoxes arise showing it is inconsistent
mathematician just invent an ad hoc add on ie axiom of seperation to save the day

are you coping DrRocket= no heart failure yet
Of course you realize that if you can show that the axioms of mathematics are inconsistent then you can also prove that

1+1 =27

is rational

and that the axioms are consistent.

In fact, with an inconsistent set of axioms you can prove ANYTHING (and also the negation of anything).

It is pretty clear that you have no idea what you are talking about. But that is natural since you are also clearly attempting to emulate Colin Leslie Dean and Dean is an idiot.

29. Well, nothing, except that since most fields of mathematics are based on it, Godel's theorem applies to them.
so what
ZFC is an ad hoc system invented to make maths consistent
it uses controversal axioms to do so
so mathematicians should not be so sure of their fields with such shaky foundations
as for godel
the theorem ie
there are maths statements which are true but cant be proven
apart from it being invalid
the theorem is meaningless
in his world and ours
as he cant tell us what makes a maths statement true

deans theorem
there are maths statements which are gibble but cant be proven
oh but dean cant tell you what a gibble statement is
i hear you laughing me out of the country
same should be done to godel

30. Originally Posted by edam421
Well, nothing, except that since most fields of mathematics are based on it, Godel's theorem applies to them.
the theorem ie
there are maths statements which are true but cant be proven
apart from it being invalid
the theorem is meaningless
in his world and ours
as he cant tell us what makes a maths statement true
Are you sure? I think that's one of the first things that is actually defined in any serious book that examines Godel's theorem. Have you read any?

31. Are you sure? I think that's one of the first things that is actually defined in any serious book that examines Godel's theorem. Have you read any?
even the cambridge expert on godel peter smith says godel did not rely on a notion of truth-thus he cant tell us what truth is or what makes a maths statement true
o
de566912ee69f0a8?lnk=gst&q=G%C3%B6del+didn%27t+rel y+on+the+notion+PETER
+smith#de 566912ee69f0a8

Quote:
Godel didn't rely on the notion
of truth

32. Of course Godel did not rely on the notion of truth in his demonstration. The point of Godel's theorem is not to explain what is true and what is not true in a given axiomatic system, he didn't need to define that in his demonstration since it had already been defined and known for decades.

33. as i said i can make 1+1=3
to be true by definition
Like me you haven,t got haven't got children yet ?
Sorry, had to make that joke but it is true that if population growth would be the context math like 1+1=3 can be meaningfull or maybe it should be 1x1=3 then (?).

What CLD is contesting is the validity of the axiom of reducibility from a philosophical point of view. This is utterly irrelevant to the conclusion of Godel's theorem, because it does not apply to everything in this world but only to those systems where this axiom is assumed to be correct. Which happens to include pretty much every piece of mathematics. Regardless of whether Russell agrees with the axiom or not, Godel's conclusions apply to every piece of mathematics ever conceived which holds the axiom of reducibility to be true.
Maybe you could keep it open that this is a possible explanation for the confusement. Godel was a mathematician and every working system in math has a certain particularity closed in itself. It,s a limit of math and that,s what Godels theorems are about.

So the axiom of reducibility is an axiom of reducability for his theorems you,re right about that. But as the whole theorem is limited for the field of math he is allowed to.

Wittgenstein was a philosopher investigating the logical form of propositions in words.

So to Wittgenstein Godel,s theorem would be a different languagegame (to use Wittgensteins ideas). Specific for math. Different game the meaning of words changes with it and as Wittgenstein believed that was not a problem as long as the meaning was clear in the context. So I,m not convinced if Wittgenstein would have a problem with the reducibility as a reducibility in the theorems. Even the word truth has no definition and doesn,t need it. So yes the theorems have a limitation as well limited to math but as Godel is clear about this there is not much of a problem I would think.

Also If Russel concludes something (or Wittgenstein it is not an argument on itself)

Russel also mentioned that "x is hot" with ice for x was false. Which to me is false if the context is not specified. As ice can have different temperatures. In relation to my skin yes it will not be hot but that is not mentioned. Ice can be hotter then ice that is colder.

"Ice is hot being false" as a proposition uses the axiom of reducability that it is experienced with our senses, so therefor "ice is hot being not true" as a general propositiont is false . But in "a certain languagegame" or use it can be true when it is clear in the use that it is about experience of the senses for instance in a biological context.

But for you,re own example 1+1=3 and the class off propositions there that becomes meaningfull (allthough I prefer 1x1=3 then) the system is no longer closed and then the criticque on the theorems would be right as the theorems would be only for a specific class of math and not math in general and that is not specified. But I have no idea if it is meant this way.

34. Of course Godel did not rely on the notion of truth in his demonstration
thus when his theorem says
there are true mathematical statements which cant be proven
and he cant tell us what a true mathematical statement is
then his theorem is meaningless

he didn't need to define that in his demonstration since it had already been defined and known for decades.
wrong
what was known for decades was the idea that maths statement was true
if it could be proved then it was considered to be true
that is why godels theorem was seen as destroying the hilbert russell attempt

http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics

n addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

he works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system
and godels theorem is meaningless as he cant tell us what makes a maths statement true

you say

Quote:
as i said i can make 1+1=3
to be true by definition
and construct a theory about gibbles

No, you cannot make 1+1=3 by definition because that would break the axiom of extensionality and you would work in an inconsistent environment.
big deal
you say godel can assume AR is correct to form his theorem -in some other world where it is correct AR is correct
even though it is invalid- in this world
then so can i assume 1+1=3 is correct -in some other world where it is correct -to prove my theorem even though it is incorrect in this worl;ld

as i said i can make 1+1=3
to be true by definition
and construct a theory about gibbles
big deal
it would be laughed out of the country
same with godel and his invalid theorem about truth -which he cant tell us what that is

you say
This is utterly irrelevant to the conclusion of Godel's theorem, because it does not apply to everything in this world but only to those systems where this axiom is assumed to be correct.
wrong
godel was not just having fun in some other world for some mere fancy
he was constructing a theorem in regard to this world
and in this world AR is invalid-even if maths uses versions of it still
and thus his theorem is invalid

35. Originally Posted by edam421
wrong
godel was not just having fun in some other world
he was constructing a theorem in regard to this world
and in this world AR is invalid-even if maths uses versions of it still
and thus his theorem is invalid
That is absurd. It sounds like something thta might come from that idiot Colin Leslie Dean.

You are confusing mathematics and physics.

Mathematics is about the implications of a set of axioms, in most caset ZFC.

Godel's theorems are as valid today as they were when his genius discovered them decades ago.

You simply fail to understand mathematics. Perhaps that comes from over-attention to writings of the idiot Colin Leslie Dean.

36. edam421 wrote:

wrong
godel was not just having fun in some other world
he was constructing a theorem in regard to this world
and in this world AR is invalid-even if maths uses versions of it still
and thus his theorem is invalid

That is absurd. It sounds like something thta might come from that idiot Colin Leslie Dean

it is said godel just assumed it true
wrong
he THOUGHT IT WAS TRUE
AND HE WAS WRONG
godel was not just having fun in some other world for some mere fancy
he was constructing a theorem in regard to this world
and in this world AR is invalid-even if maths uses versions of it still
and thus his theorem is invalid

further

Quote:
he didn't need to define that in his demonstration since it had already been defined and known for decades.

wrong
what was known for decades was the idea that maths statement was true
if it could be proved then it was considered to be true
that is why godels theorem was seen as destroying the hilbert russell attempt

http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics

Quote:
n addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

he works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

and godels theorem is meaningless as he cant tell us what makes a maths statement true

37. We are reading the theorem different on this part :

statements that are true but cannot be proven within the system
To you and maybe Mr Dean this implies automatically the axiom of reducibility.

But to me this doesn,t and I can explain why.

As you know according to wittgenstein Meaning of symbols or language is just use.

Example of a thought experiment :

Wittgenstein asks the reader to perform a thought experiment: to come up with a definition of the word "game".[6] While this may at first seem a simple task, he then goes on to lead us through the problems with each of the possible definitions of the word "game". Any definition which focuses on amusement leaves us unsatisfied since the feelings experienced by a world class chess player are very different from those of a circle of children playing Duck Duck Goose. Any definition which focuses on competition will fail to explain the game of catch, or the game of solitaire. And a definition of the word "game" which focuses on rules will fall on similar difficulties.
You see the meaning of a word adapts to the context it is part of. And in the original language it is more clear as the german word "spiele" also involves play.

Beit a game of solitaire or a game of chess the meaning changes with the context different context or system does not mean a different world, all these games are part of live.

Here the context for one sentence where the word is used mostly is the same.

But in Godels theorem it is not (at least not how I read it and you should proove that I read it wrong which is difficult as we can,t ask Godell for it)

The word prooven implies a truth ; "a prooven truth".

That would mean it implies the axiom of reducibility I agree to that. But the way I read it the context for the word true and the word prooven is not the same.

Prooven is meant as "not a prooven truth in general" (where the world or live as a whole is or would be the context) and true the context is just the system, the particular use. Now within the system containing arythmatics that are true for the system these truths can,t be prooven when you change the context from the system to real live as then the system stops being a closed system.

So the theorem not only not implies the theorem of reducibility it even rejects it.

The language game as Wittgenstein uses that notion (context use and reference) is different within one sentence for the words prooven and truth.

If you read it for the same you have the theorem of reducibility if you read it as I do it rejekts that theorem. So how do we have to read it ? If dean critisizes the theorem when it is read as you (and he himself maybe) seem to do his criticque is runderstandable but it is a criticque that has a certain reading as theorem of reducibility for both sides of the discussion then. And I think Godell might feel misunderstood by both sides of a discussion then for reading it that way.

For instance two people read it like as it implies the axiom of reducibility then they have a discussion about the theorem being false or not. One holds an axiom for reducibility as not a problem and the other does rejekt the theorems as it isa problem. I hold it possible Godell could feel his theorems misinterpreted by both.

But maybe the most by those that feel an axiom of reducibility is not a problem according to the theorem. As the theorem seems to rejekt such axioms the way I (and AlexandruLazar) read it. Now you have to proof that we read it false and as Godell doesn,t live anymore that is a difficult and a bit senseless discussion.

38. This is utterly irrelevant to the conclusion of Godel's theorem, because it does not apply to everything in this world but only to those systems where this axiom is assumed to be correct.
wrong
godel was not just having fun in some other world for some mere fancy
he was constructing a theorem in regard to this world
and in this world AR is invalid-even if maths uses versions of it still
and thus his theorem is invalid
That's the whole goddamn point, he was not dealing with this world! He was not constructing a theorem in regard to this world, he was constructing a theorem in regard to a particular class of axiomatic systems which just so happens to include most branches of mathematics.

Besides that, you do not understand what an axiom is.

you say

Quote:
as i said i can make 1+1=3
to be true by definition
and construct a theory about gibbles

No, you cannot make 1+1=3 by definition because that would break the axiom of extensionality and you would work in an inconsistent environment.
big deal
you say godel can assume AR is correct to form his theorem -in some other world where it is correct AR is correct
even though it is invalid- in this world
then so can i assume 1+1=3 is correct -in some other world where it is correct -to prove my theorem ev[qen though it is incorrect in this worl;ld
No, you cannot assume 1+1=3 is correct because this axiom breaks another axiom, namely the axiom of extensionality. Godel can freely assume that the axiom of reducibility is correct because it does not contradict any other axiom in his system. Yours does, unless you admit that the axiom of extensionality is wrong -- i.e. unless you want to say that two sets can be equal even if they do not have the same elements.

Originally Posted by edam421
Of course Godel did not rely on the notion of truth in his demonstration
thus when his theorem says
there are true mathematical statements which cant be proven
and he cant tell us what a true mathematical statement is
then his theorem is meaningless
No, it is not. And it seems like you did not even bother to read the quotes you have copied and pasted . I won't bother you with formulas and formal definitions, it basically boils down to the fact that a statement is simply defined to be true if the form associated with it holds for every case in which it is defined; for any particular case you may of course test whether the statement is true or not, but it may not be possible to actually prove whether it stands or not for every possible case. A simplistic example is a conjecture somewhere along the lines of "X is true for any natural number". It is obviously possible to verify whether X is true for any particular number you pick at random; however, it may not be possible to prove that X is true for any natural number. The statement itself may well be true for any possible number -- except that it might, in fact, be impossible to prove so. This is the simplest situation (where it is possible to verify the form for a given case, but you cannot prove its generality); there are other, more complicated situations though.

This is not some modern development of mathematics that came after Godel did his work -- it's something that was actually known at the time. Turing and Church used the notion of mathematical truth in a different context, as they were more concerned with other properties than those dealt with by Godel. Their work was not concerned with the ontological truth (i.e. the intrinsical property of actually being true) but with the decidable truth (i.e. the property of being known to be true) which was relevant to their work.

39. you said

Godel's theorem, because it does not apply to everything in this world but only to those systems where this axiom is assumed to be correct
wrong
as i said godel is not assuming for the sake of some logic excercise that AR is valid in some nether world
he is doing a proof which he thinks is relevant to this world
and he regards AR as valid in this world
and in this world AR is invalid as many have pointed out

thus his proof/theorem is not valid in this world

edam421 wrote:
Quote:
Of course Godel did not rely on the notion of truth in his demonstration

thus when his theorem says
there are true mathematical statements which cant be proven
and he cant tell us what a true mathematical statement is
then his theorem is meaningless

No, it is not. And it seems like you did not even bother to read the quotes you have copied and pasted Smile. I won't bother you with formulas and formal definitions, it basically boils down to the fact that a statement is simply defined to be true if the form associated with it holds for every case in which it is defined;

i see you ignored the point that for decades mathematicians regarded true as provability
and that godel is said to have demolished that-with out telling us what truth is thus his theorem is meaningless

what was known for decades was the idea that maths statement was true
if it could be proved then it was considered to be true

that is why godels theorem was seen as destroying the hilbert russell attempt

http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics

Quote:
n addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

he works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

and godels theorem is meaningless as he cant tell us what makes a maths statement true

IF YOU ASKED GODEL AT THE TIME OF HIS PROOF
WHAT MAKES A MATHEMATICAL STATEMENT
WHAT WOULD HE HAVE SAID

No, you cannot assume 1+1=3 is correct because this axiom breaks another axiom, namely the axiom of extensionality. Godel can freely assume that the axiom of reducibility is correct because it does not contradict any other axiom in his system. Yours does,
sorry in my artifical world 1+1=3 and the axiom of extensionality does not exist there
just as godel can use AR in his artifical world

but in our world it is invalid

40. Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves
Here you are also interpreting.
ou put you,re definition of a word to the word you read.

I,m not sure but a constructed proposition or just a proposition seems not the same to me. Even a single number can be a proposition for instance in relation to an amount of things. Being a symbol without an idea about what it symbolizes it is meaningless. But constructed it says something if even if alone about how it is constructed. The parts within the proposition can be compared with each other also. 3/1 can give an idea about a relation even if you have no idea what the nubers stand for. They cant say something about themselves but they can about their relation to other parts of the proposition.

41. Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drasti]... We saw that we can construct propositions which make statements about themselves

Here you are also interpreting.
ou put you,re definition of a word to the word you read.
no interpreting
godel is clearly saying he is construtiing propositions which russell etc said were invalid
its godels word not mine

42. If you read you interpret every single word let alone combinations and the total of words.

He is saying that, if you read that literally, but the proposition that is construkted will automatically be a "constructed proposition" and not any proposition, he leaves out all propositions that are not constructed and all that are constructed but do not say anything about themselve.

An example of a constructed proposition saying something about itself in math is a magic-scquare. The construktion says something about itself even if you don,t know what each number stands for.

43. Originally Posted by edam421
he is doing a proof which he thinks is relevant to this world
For the I-don-t-know-how-many-th time, he is doing a proof in a formal system which is a world of its own. It just so happens that (ironically, unlike Wittgenstein and Russell's system!) some elements from this world can be modeled within that system, but they are completely disjoint.

thus his proof/theorem is not valid in this world
Of course it is not, because this world is not an axiomatic closed system. Unfortunately, this means that Wittgenstein's arguments are not true in this world, either, and neither are Russell's, and neither are those of the blockhead you keep quoting.

i see you ignored the point that for decades mathematicians regarded true as provability
and that godel is said to have demolished that-with out telling us what truth is thus his theorem is meaningless
I see you ignored the point that when Godel developed his theorem, mathematicians had been making a distinction between ontological and decisional truth for decades. What they were assuming (and Godel had proved to be false) was that anything that is ontologically true can also be proven.

IF YOU ASKED GODEL AT THE TIME OF HIS PROOF
WHAT MAKES A MATHEMATICAL STATEMENT
WHAT WOULD HE HAVE SAID
Exactly what I said above. That a statement is said to be true if the predicate associated with it holds in every case in which it is defined. This is exactly what Turing, Church, Russell, or hell, Leibniz, Newton or Thales himself would have said. The only thing that Godel showed was that a predicate may hold for every case in which it is defined, but it may be impossible to show it formally -- which sounds awfully simple but isn't too straightforward.

sorry in my artifical world 1+1=3 and the axiom of extensionality does not exist there
If the axiom of extensionality does not exist in your world, you cannot have uniquely determined natural numbers, so 1+1=3 is actually meaningless (not accepting the axiom of extensionality means that any two numbers can be equal and unequal at the same time). Not accepting AE introduces an inconsistency in your system; ironically, introducing AR does not.

44. For the I-don-t-know-how-many-th time, he is doing a proof in a formal system which is a world of its own. It just so happens that (ironically, unlike Wittgenstein and Russell's system!) some elements from this world can be modeled within that system, but they are completely disjoint.

Quote:
thus his proof/theorem is not valid in this world

godel did not think AR was invalid in this world
he thought it was valid

godel is doing a proof /theorem which he thinks will be valid in this world
as i said godel thought AR was valid and it is not
and he thought his proof/theorem valid in this world
but it is not as in this world AR is invalid

as i said godel is not assuming for the sake of some logic excercise that AR is valid in some nether world
he is doing a proof which he thinks is relevant to this world
and he regards AR as valid in this world
and in this world AR is invalid as many have pointed out
I see you ignored the point that when Godel developed his theorem, mathematicians had been making a distinction between ontological and decisional truth for decades. What they were assuming (and Godel had proved to be false) was that anything that is ontologically true can also be proven.

Quote:
IF YOU ASKED GODEL AT THE TIME OF HIS PROOF
WHAT MAKES A MATHEMATICAL STATEMENT
WHAT WOULD HE HAVE SAID

Exactly what I said above. That a statement is said to be true if the predicate associated with it holds in every case in which it is defined. This is exactly what Turing, Church, Russell, or hell, Leibniz, Newton or Thales himself would have said. The only thing that Godel showed was that a predicate may hold for every case in which it is defined, but it may be impossible to show it formally -- which sounds awfully simple but isn't too straightforward.

What they were assuming (and Godel had proved to be false) was that anything that is ontologically true can also be proven
what godel and mathematicians of his time regarded as truth was provability
as the wiki reference shows

Quote:
n addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.
and godel cant tell us what a true maths statement is -as he like the rest believed truth was provability-which he is said to have disproved
thus theorem meaningless

IF YOU ASKED GODEL AT THE TIME OF HIS PROOF
WHAT MAKES A MATHEMATICAL STATEMENT
WHAT WOULD HE HAVE SAID
he would have said he did not know

you say
That a statement is said to be true if the predicate associated with it holds in every case in which it is defined. This is exactly what Turing, Church, Russell,
as wiki quote shows

turing church russell all regarded truth as provability
russell contructed PM based on the idea that truth was provability

sorry in my artifical world 1+1=3 and the axiom of extensionality does not exist there

If the axiom of extensionality does not exist in your world, you cannot have uniquely determined natural numbers, so 1+1=3 is actually meaningless (not accepting the axiom of extensionality means that any two numbers can be equal and unequal at the same time). Not accepting AE introduces an inconsistency in your system; ironically, introducing AR does not.
so what
i can construct any system i like for an intellectual excersise in my nether world
just as you allow godel to do with the invalid AR
you say the inconsistency can be avoided by introducing AR
in my system i can do what i want
ie making 1+1=3 and useinhg AR
both assumed to be valid in my nether world
but both invalid in this world

there are maths statements which are gibble but cant be proven
but i cant tell you what a gibble statement is
i hear you laughing me out of the country
same should be done to godel
[/quote]

45. it is shown by colin leslie dean that Godels first theorem ends in paradox

it is said godel PROVED
"there are mathematical true statements which cant be proven"
in other words
truth does not equate with proof.

if that theorem is true
then his theorem is false

PROOF
for if the theorem is true
then truth does equate with proof- as he has given proof of a true statement
but his theorem says
truth does not equate with proof.

46. A system is never another world or seperate from this world. Like a game of chess or sollitaire it is not a world on it,s own.

You could say a game of chess has a certain closedness but it is not closed. Music and art are not seperate worlds also. You can see the world as a manifold of closed systems and use math on it. If the math is true in these cases but the idea of closed systems isn,t then the math is not generating truths. Not even for a "system" it is used for. Math then in the way it is used projekts the situation as being a closed system and isolated. So the mathematical statements can be true but only thanks to the isolation from this world on the paper creating a closed system.

Regarding something as a closed system while it is created (by doing the math) to be a closed system is false. Then the math is a closed system and that is projekted on a part of reality and thus a part of this world.

So Where am I in this discussion ? No idea.

By the way the symbol = I have learned to write as => at highschool (highschool in european sense not university).

Not as a symbol for ecqual thus. Where I see the symbol = in math I don,t read it necessarily as ecqual but as a logic step or result.

Also in most math of the form A+B=C it is assumed that the numbers stand for a similar thing. That can be apples, pears, fruit or things but not A for apples and B for pears. If A would be apples and B pears A and B (and C) can stand for fruit.

But the assumption is a certain category. So 2 pears + 2 apples is possible if you generalize to fruit.

In the case of 1+1=3, 1 and 1 is for the category of a species but 1 is not ecqual to 1 as 1 is male and 1 is female. Then as I use the symbols = and => through each other (in this case => would be more appropriate) the axiom of extensionality is not a problem. I explained the way I used the math at forehand.

The problem is that the category for the numbers is not the same.
For instance if the category would be mice. 1+1=3 could also count for two male mouses. And with a male and female it also would have to count always which is also not true. Therefor I mentioned 1x 1 =3 as more appropriate.

Because in statements of the form AxB=C in contrast with A+B=C A and B can,t be similar category. A scquared apple is nonsense. A can be for apples or B but not both.
For instance for a tilefloor of 6 x6 you have not 30 scquared tiles. there is a perpendicularity that makes that 6 does not ecqual 6. Put all the 36 tiles in one row and you see that it is different. 1x36 does not ecqual 6x6 in this case allthough both cases the amount of tiles is the same.

47. edam421, I can see your problem is that you are actually unable to even read the quotes you are copying and pasting. I'm sorry to see you have picked Mathematics as your field of interest, you could have been a famous Catholic minister.

48. Originally Posted by edam421
it is shown by colin leslie dean that Godels theorem ends in paradox

it is said godel PROVED
"there are mathematical true statements which cant be proven"
in other words
truth does not equate with proof.

if that theorem is true
then his theorem is false

PROOF
for if the theorem is true
then truth does equate with proof- as he has given proof of a true statement
but his theorem says
truth does not equate with proof.
Wrong.

The existence of a true but unprovable theorem does not imply that Godel's theorem is an example of such. There are also theorems that are true AND provable.

Your reasosning shows lack of logic, which is what one expects from Colin Leslie Dean. Colin Leslie Dean is an idiot. Emulating Colin Leslie Dean is idiotic.

49. CLD's demonstration is flawed. What does 'true' mean? What does 'proof' mean?

50. Godel's theorem has been misstated. The correct statement of the theorem is that in any logical system, which is strong enough to do arithmetic, there are statements which are unprovable, i.e. cannot be proven true or false.

51. Originally Posted by mathman
Godel's theorem has been misstated. The correct statement of the theorem is that in any logical system, which is strong enough to do arithmetic, there are statements which are unprovable, i.e. cannot be proven true or false.
No, the statement made earlier is basically correct. There are true statements that are not provable within the system.

The proof goes by construction of a sentence, called now a Gödel sentence, that cannot be proved true within the system but is shown to be true by a metamathematical argument.

Basically the proof is effected by using formal logic to write the sentence, call it S, "This sentence is not provable", within the rules of formal logic.

Now assume that S is is not true. Then simply because of what S is, it is in fact true. So we have that S is true. It is therefore not provable. But it is true.

Now, this results in your assertion that there are undecidable statements. In fact it would be completely uninteresting if only false statements were unprovable. Since given any valid sentence A, either A must be true or -A must be true. And proving either one would establish the validity of the other. So the statement that there are true but unprovable statements is not really a restriction, since if A is unprovable so s -A.

The theorem is called an incompleteness theorem precisely because there are true statements that are not provable. A complete system is one in which all true statements are provable. A consistent system is one in which only true statements are provable, or equivalently one in which no statement can be proved to be both true and false.

Warning: All of the above is stated in more or less "plain English". You can get yourself confused and turned around in the semantics, This problem does not occur if you stick to formal symbolic logic, but that becomes rather pedantic.

You might want to take a look at the link I posted earlier (in one of these threads on Gödel) to a translation of Gödel's original paper.

Gödel's second theorem is that the assertion that the axioms are consistent is undecidable.

There are metamathematical proofs, I am not familiar with the details, of the consistency of the axioms of arithmetic.

52. There are also theorems that are true AND provable.
fact is truth is not equated with proof
poof of a statement cannot be in any way since godel a be regarded as
makeing a statement true

fact is truth is not equated with proof
poof of a statement cannot be in any way since godel a be regarded
as makeing a statement true

true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

he works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

53. edam421, I can see your problem is that you are actually unable to even read the quotes you are copying and pasting. I'm sorry to see you have picked Mathematics as your field of interest, you could have been a famous Catholic minister.
yes i conced i may have been wrong about
turing church .. all regarded truth as provability
but
russell contructed PM based on the idea that truth was provability
when godel is said to have shown truth was not provability
he did not tell us what then made a maths statement true

you could have been a famous Catholic minister.
i could not as everything
scientisim
science
maths
religion
all end in meaninglessness

54. colin leslie dean has shown Godels second theorem ends in paradox

http://en.wikipedia.org/wiki/G%C3%B6...second_theorem
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
now this theorem ends in self-contradiction

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done

55. edam421, I see you are ignoring my question. In CLD's system, how are 'truth' and 'proof' defined?

56. he did not tell us what then made a maths statement true
Yes, he did, that's the entire point. Throughout the demonstration, it is implied that a statement is considered to be true if it holds for every case in which it is defined, i.e. if it is consistent with the system's axioms. To CLD's defense, the definition is indeed not stated explicitly in the theorem's demonstration, but this is really reasonable for anyone with fully-developed mental capacity, considering for instance that Pythagoras didn't bother to explicitly state what "equal" means when he said that the sum of the squares of legs equals the square of the hypothenuse. Are you implying his theorem ends in contradiction as well?

In addition, most textbooks on the subject usually explain this distinction between truth in the sense of provability and truth in the sense of consistency, exactly in order to clear up this misunderstanding which is not uncommon among math students. Unfortunately, it's quite clear that CLD didn't bother to read any actual textbook on the subject (or read a poor one).

57. Originally Posted by AlexandruLazar
he did not tell us what then made a maths statement true
Yes, he did, that's the entire point. Throughout the demonstration, it is implied that a statement is considered to be true if it holds for every case in which it is defined, i.e. if it is consistent with the system's axioms.
Gödel had no need to define the meaning of "true". He was working in the context of formal logic and "truth" goes back to Aristotle's original development of formal logic and truth tables.

It is semantically consistent with your explanation.

Originally Posted by AlexandruLazar
To CLD's defense, the definition is indeed not stated explicitly in the theorem's demonstration, but this is really reasonable for anyone with fully-developed mental capacity, considering for instance that Pythagoras didn't bother to explicitly state what "equal" means...
There is no defense for CLD. He is an idiot.

See earlier note regarding truth in formal logic, which is the context in which Gödel was working, and in fact the context for all of mathematics at the most fundamental level.

Originally Posted by AlexandruLazar
In addition, most textbooks on the subject usually explain this distinction between truth in the sense of provability and truth in the sense of consistency, exactly in order to clear up this misunderstanding which is not uncommon among math students. Unfortunately, it's quite clear that CLD didn't bother to read any actual textbook on the subject (or read a poor one).
Better be careful here. "Consistency" is usually used to refer to a set of axioms and means something rather different. A consistent set of axioms is one in which no false statement is provable, i.e. one in which the axioms do not contradict themselves. Once you have an inconsistent set of axioms everything goes haywire, because anything, literally anything, can be shown to be both true and false.

CLD is so far out in left field that there is no explaining him in rational terms, as is the case in general with babbling idiots.

58. Originally Posted by AlexandruLazar
edam421, I see you are ignoring my question. In CLD's system, how are 'truth' and 'proof' defined?
Whatever give you the idea that CLD has a system ?

A system is generally the product of rational thought processes. CLD has displayed no such characteristic.

It is not possible to have a rational discussion with an irrational person, and though CLD is not here, it is his writings that have been used to argue against Gödel and by proxy all of mathematics.

59. Indeed; there is a difficulty for me here because I am not entirely familiar with the English terminology, as my contacts with this field have taken place in my native language which is not English.

When I say that a statement must be consistent within a given system, I mean that it is in accordance with the axioms of that system. For instance, in the context of ZFC it can be shown that 2=3 is a false statement because it is not in accordance with the axiom of extensionality, which says that two sets are equal if they consist of the same elements. I'm not sure if the use of 'consistent' in this context is correct in English -- I hope I'm not confusing anyone.

Edit: of course there's no defense for CLD, but this is not the first time I see this kind of confusion. Turing, for instance, explicitly uses the similarity between provable and true, but not because this is something he considers to be a fundamental definition of mathematics (i.e. the meaning of 'true') but because he is working in the context of analyzing a finite automata.

60. I'm pretty sure CLD doesn't have a coherent system or anything besides this bombastic statement in the field of mathematics -- it's exactly why I asked what those definitions are.

61. Originally Posted by edam421
colin leslie dean has shown Godels seconded theorem ends in paradox

http://en.wikipedia.org/wiki/G%C3%B6...second_theorem
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
now this theorem ends in self-contradiction

Not if you understand what an inconsistent set of axioms is. Apparently you do not.

In an inconsistent set of axioms one can prove ANYTHING. That includes any sentence and the negation of that sentence. ANYTHING. This simply because a false premise implies anything -- a simple fact from ordinary Aristotelian logic.

So it is no surprise that in an inconsistent system you can prove that the system is consistent. You can also prove that it is inconsistent. You can also prove that it is orange.

The content of the Gödel theorem is that if the axioms are sufficiently rich as to permit ordinary arithmetic, and if they are are consistent, then the consistency is not provable within the set of axioms. This does not exclude metamathematical proofs of consistency.

Colin Leslie Dean did show that Gödel's theorem ends in paradox. Colin Leslie Dean only proved that he does not understand the Gödel theorems. Colin Leslie Dean is an idiot.

62. Originally Posted by AlexandruLazar
I'm pretty sure CLD doesn't have a coherent system or anything besides this bombastic statement in the field of mathematics -- it's exactly why I asked what those definitions are.
It was a rhetorical question. See last sentence in earlier post.

63. Originally Posted by AlexandruLazar
Indeed; there is a difficulty for me here because I am not entirely familiar with the English terminology, as my contacts with this field have taken place in my native language which is not English.

When I say that a statement must be consistent within a given system, I mean that it is in accordance with the axioms of that system. For instance, in the context of ZFC it can be shown that 2=3 is a false statement because it is not in accordance with the axiom of extensionality, which says that two sets are equal if they consist of the same elements. I'm not sure if the use of 'consistent' in this context is correct in English -- I hope I'm not confusing anyone.
Your use of the term consistent is correct in the English language. Your English is very good.

The problem lies with the use of special terms in mathematics, and in particular in formal logic. In formal logic there is a very specific meaning to a "consistent set of axioms", which is germane to the topic at hand.

64. Gödel had no need to define the meaning of "true". He was working in the context of formal logic and "truth" goes back to Aristotle's original development of formal logic and truth tables.

true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

shook this assumption, with the development of statements that are true but cannot be proven within the system

godel is said to have destroyed the notion that truth as provability
proof of a statement has nothing to do with it being true -but godel cant tell us what makes a maths statement true thus his theorem is meaningless

http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

proof of a statement has nothing to do with it being true -but godel cant tell us what makes a maths statement true thus his theorem is meaningless

65. Not if you understand what an inconsistent set of axioms is. Apparently you do not.
But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent .

. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done

66. You are the one who cannot read. I already explained to you several times that Godel could tell us perfectly well what a true statement was.

67. You are the one who cannot read. I already explained to you several times that Godel could tell us perfectly well what a true statement was.
the historical fact is
at the time of godels proof what was regarded as being true was provability

true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system

68. For the one thousand-th time, I am not talking about what was generally regarded as true but about what Godel himself regarded as true. There was a time when the Earth was generally regarded as flat, this doesn't mean some people didn't think it was round.

Also, I see that despite having carried this discussion over three or four threads, you are still disregarding my question about how CLD defines 'true' and 'provable'. Does he have a definition for these? If he doesn't, then by his own argument his demonstration is wrong.

69. I haven't heard of this CLD character until a few days ago but from this argumentation I begin to think he is an idiot as Dr. Rocket says. This is the really one of the most goddamn basic things about mathematical logic and it is within the first few pages of the first chapter of even the most poorly-written textbook,

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
The system of mathematical logic in which Godel did his demonstration cannot be proven to be both consistent and complete from within itself. Therefore, by Godel's theorem, it is not inconsistent. If I remember well, it has been proven to be consistent (which is not too difficult), but it cannot be proven to be complete from within itself.

70. Originally Posted by AlexandruLazar
I haven't heard of this CLD character until a few days ago but from this argumentation I begin to think he is an idiot as Dr. Rocket says. This is the really one of the most goddamn basic things about mathematical logic and it is within the first few pages of the first chapter of even the most poorly-written textbook,

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
The system of mathematical logic in which Godel did his demonstration cannot be proven to be both consistent and complete from within itself. Therefore, by Godel's theorem, it is not inconsistent. If I remember well, it has been proven to be consistent (which is not too difficult), but it cannot be proven to be complete from within itself.
That is not quite correct.

What Gödel showed is that any system of axioms that admits the natural numbers cannot be proved, within the system to be consistent. There are, as I recall, proofs of the completeness of Peano arithmetic, in larger axiom systems, but then one cannot prove the larger set of axioms to be complete within that set of axioms. I also recall that there are metamathematical proofs of the consistency of Peano arithmetic. These proofs are, as I recall, quite difficult. That is to be expected as proof of the consistency of the axioms of arithmetic was one of the original Hilbert problems and there are no easy Hilbert problems.

Next, assume the axioms to be consistent. There is no evidence that they are not, and there is the metamathematical proof (which post-dates Gödel's theorems). Also, as noted earlier there is no point in working with an inconsistent set of axioms.

If the axioms are consistent, then they are incomplete -- which means that there are true but unprovable theorems.

The second of Gödel's theorems is that, again assuming that the axioms are consistent, that this fact is not provable within the set of axioms. If, on the other hand the axioms are inconsistent, then it is possible that there does exist a proof that they are consistent (recall that in an inconsistent system, any statement including the negation of any statement is provable).

Nobody, at least nobody who understands mathematics, believes that the axioms are inconsistent. That would result in just too mahy absurd conslusions, including that 1+1=3 (or 1+1 = for that matter).

71. Originally Posted by AlexandruLazar
Also, I see that despite having carried this discussion over three or four threads, you are still disregarding my question about how CLD defines 'true' and 'provable'. Does he have a definition for these? If he doesn't, then by his own argument his demonstration is wrong.
You need to recognize that the benefit of your posts will be realized by lurkers, likely young folks and not by the OP.

There is no hope for the OP. Remember that he is basing his position on the work of Colin Leslie Dean, a demonstrable idiot, and he who deliberately emulates an idiot is beyond help.

Remember that Colin Leslie Dean, among his several claims, is the self-proclaimed leading erotic poet of Australia. Apparently only Dean and the OP can see the connection to deep expertise in symbolic logic. They seem to see a lot of things. Most of us would call those things hallucinations.

It is reasonably clear to the casual observer, that whatever might be the system of Colin Leslie Dean, it is inconsistent. This is abundantly clear from the observation that he manages to "prove" so many things that are patently false. In such a system neither "truth" nor "provability" are meaningful.

All of these threads on Gödel theorems are utter rubbish and belong in Pseudoscience. They do not belong in the mathematics forum.

72. What Gödel showed is that any system of axioms that admits the natural numbers cannot be proved, within the system to be consistent
godel showed

http://en.wikipedia.org/wiki/G%C3%B6...second_theorem
Quote:
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done

73. Originally Posted by edam421
What Gödel showed is that any system of axioms that admits the natural numbers cannot be proved, within the system to be consistent
godel showed

http://en.wikipedia.org/wiki/G%C3%B6...second_theorem
Quote:
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done
wrong

74. Also, I see that despite having carried this discussion over three or four threads, you are still disregarding my question about how CLD defines 'true' and 'provable'. Does he have a definition for these? If he doesn't, then by his own argument his demonstration is wrong.
its not about what dean can do
fact is godel canot tell us what makes a maths statement true
thus his theorem is meaningless

75. Originally Posted by edam421
Also, I see that despite having carried this discussion over three or four threads, you are still disregarding my question about how CLD defines 'true' and 'provable'. Does he have a definition for these? If he doesn't, then by his own argument his demonstration is wrong.
its not about what dean can do
fact is godel canot tell us what makes a maths statement true
thus his theorem is meaningless
1) I would be happy to tell you what Dean can do. It might be considered anatomically difficult.

2) Gödel's theorem is quite meaningful to those with the knowledge and intelligence to understand it. That, of course, rather leaves Dean out in the cold.

76. wrong

godel showed

Quote:
http://en.wikipedia.org/wiki/G%C3%B6...second_theorem
Quote:
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
Quote:
But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . ... But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done

77. @DrRocket: No, that particular rephrasing is correct, it's just that the logic system in which Godel demonstrated the theorem cannot be proven to be complete from within itself. Therefore, there is no paradox -- it is not an axiomatic system that can be proven to be both consistent and complete from within itself, so as to be inconsistent per his own theorem.

78. Originally Posted by edam421
Also, I see that despite having carried this discussion over three or four threads, you are still disregarding my question about how CLD defines 'true' and 'provable'. Does he have a definition for these? If he doesn't, then by his own argument his demonstration is wrong.
its not about what dean can do
fact is godel canot tell us what makes a maths statement true
thus his theorem is meaningless
Of course it is about what dean can do.

You are saying that CLD has shown Godel's theorem to be inconsistent. Does CLD have a definition for what is "true", "provable" and "consistent"? If not, by his own arguments, such a proof is bogus -- he says that Godel's theorem is inconsistent, but he doesn't tell us what inconsistent is.

I'm just applying CLD's logic -- which is incorrect in fact, but if you insist on thinking it is correct, that's ok, it may be easier to show CLD's idea is bogus by resorting to his own demonstration, rather than notions about mathematics that you (and quite possibly him) clearly do not understand.

79. @DrRocket: No, that particular rephrasing is correct, it's just that the logic system in which Godel demonstrated the theorem cannot be proven to be complete from within itself. Therefore, there is no paradox -- it is not an axiomatic system that can be proven to be both consistent and complete from within itself, so as to be inconsistent per his own theorem.
thus

a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . ... But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done
if his logic is inconsistent
then he could prove anything ie the opposite of his theorem
thus again his theorem would be meaningless

80. The content of the Gödel theorem is that if the axioms are sufficiently rich as to permit ordinary arithmetic, and if they are are consistent, then the consistency is not provable within the set of axioms. This does not exclude metamathematical proofs of consistency.
This is also an interpretatoig and in relation to this interpretation I can even understand Edam as well as Dean,s critic to Godell. But if they react against this type of interpretation then they should do that and not interpret it the same way.

This way of interpreting (at least I can,t imagine or am not sure yet Godell meant this) has a risk.

A mathematical system can,t be proven consistent by itself. But I suppose it neither can be proven inconsistent then ? Lack of proovability would offcourse also imply that.

So we actually can know (as Dean argues) nothing exept that the mathematical system is consistent in itself. But is impossible to proove or disproof that so it makes no sence trying to argue against the set of axioms. The system works on the axioms and as the system can't be prooven inconsistent with reality the axioms, prooved by the consistency of the mathematical system cannot be prooven false either. the system we should except it because of it,s mathematical consistency ? That,s methafysics. Any system that has some consistency in itself can defend itself by this.

"Consistency is not proovable" means "there is no proovability of consistency ór inconsistency", no proovability whatsoever at least, not in the mathematical consistency.

What is consistency anyway ? You can build a perfectly consistent logic system based on cirkle logic. Consistent but nevertheless circle-logic so not relevant.

Also consistency alo says not much about completeness or incompleteness, how much it is simplified.

81. Originally Posted by edam421
@DrRocket: No, that particular rephrasing is correct, it's just that the logic system in which Godel demonstrated the theorem cannot be proven to be complete from within itself. Therefore, there is no paradox -- it is not an axiomatic system that can be proven to be both consistent and complete from within itself, so as to be inconsistent per his own theorem.
thus

a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . ... But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done
if his logic is inconsistent
then he could prove anything ie the opposite of his theorem
thus again his theorem would be meaningless
Have you even read what I wrote? I said logic cannot be proven to be complete from within itself. It is perfectly consistent, but cannot be proven to be complete. By that rephrasing, it should be proven to be both consistent and complete, from within itself, to be inconsistent. As it cannot be proven to be complete from within itself, it is not inconsistent by Godel's theorem.

---

Now for someone with meaningful issues.

A mathematical system can,t be proven consistent by itself. But I suppose it neither can be proven inconsistent then ? Lack of proovability would offcourse also imply that.
No, not really. In some cases, it may be possible to prove that a system is inconsistent. For instance, a system based on the following two axioms:

A: Given a straight line l and a point exterior to it P, one can only pass a single straight line m through it so that l and m are parallel

B: Given two points P and Q, one can pass an infinity of distinct straight lines through them

can be proven to be inconsistent from within itself. To do so, just suppose you take the straight line l and a point exterior to it P. Now take another point Q so that a line that passes through both P and Q is parallel to l. By axiom A, this should be the only line that is parallel to l and passes through Q, but by axiom B, you can pass an infinity of distinct lines through P and Q, which would all be parallel to l. This contradicts A.

Consistency is far more difficult to prove than inconsistency, as inconsistency can be proven through a single example, whereas consistency has to be addressed by other means.

"Consistency is not proovable" means "there is no proovability of consistency ór inconsistency", no proovability whatsoever at least, not in the mathematical consistency.
I think this is already answered by what I said above, but in any case, just to clear up this issue: "Consistency is not provable" is not rephraseable as "there is no provability for consistency or inconsistency". It may be possible to prove a system to be inconsistent (and if it's bad enough, that's fairly straightforward), but proving that it is consistent (or, equivalently, that it is not inconsistent) is far more difficult.

Ironically, Godel's work was actually based quite heavily on axiomatic set theory which appeared exactly because Hilbert's set theory was shown to be inconsistent (just google Russell paradox for that).

Also, please remember that the interpretation given above requires that the system be proven to be consistent and complete from within itself.

What is consistency anyway ? You can build a perfectly consistent logic system based on cirkle logic. Consistent but nevertheless circle-logic so not relevant.
No, actually you can't, because circular logic does not allow you to evaluate if a statement is true or false. It doesn't make sense to speak of consistency or inconsistency in this case -- you cannot end up showing that a statement is shown to be true by one axiom and false by another one because you can't even end up showing that it is true or false by any of them.

P.S. Just for the sake of showing how subtle mistakes can slip in unsuspected, the mistake you are doing in:

"Consistency is not proovable" means "there is no proovability of consistency ór inconsistency", no proovability whatsoever at least, not in the mathematical consistency.
is assuming that not being able to prove a statement also means you cannot prove its negation. This isn't always true. To put it in basic logic terms and apply it to our case, you may be unable to prove the statement "P is (in the set of) Q" because you may not be able to show that P has all the properties required to be in the set of Q (i.e. that a system (P) has all the properties required to be in the set of consistent systems (Q)) . On the other hand, "P is not (in the set of) Q" can be proven simply by showing that there is one property required for P to be in Q which P does not have.

82. I said logic cannot be proven to be complete from within itself.
you just dont get it

the point is
godel has created a system within which he proves his theorem

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

83. Look, let's put it this way. Godel's second theorem can be rephrased, by your quote, as follows:

If a system can

A. Be proven to be consistent from within itself AND
B. Be proven to be complete from within itself

then it is inconsistent.

It is your quote, word by word, here it is for reference:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
CLD is trying to show that logic can be proven to be consistent from within itself, and as per Godel's theorem, it must be inconsistent. However, this is not what Godel's theorem says (which you can see right above: the system must meet both A and B, not just A).

Godel's logic does not meet the criteria B. I'm not even sure if it meets A, but let's assume CLD is right (which is unlikely, but what the heck) and it does. It still does not meet B. There is no paradox here, as CLD implies. He is not applying the theorem correctly. His theorem does not imply that a system cannot be shown to be consistent. As long as it can be shown it is consistent, but not that it is complete (or the other way round), everything is fine.

84. Look, let's put it this way. Godel's second theorem can be rephrased, by your quote, as follows:
look
godel is developing a proof with in a system of mathematics
which he takes to give a true theorem

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

85. colin leslie dean points out The fundamental problem with Godels theorems
are he creates an imprdeicative statement and the theorems apply to themselves ie are impredicative- thus leading to paradox

ieie this is godels impredicative statement used in his first theorem
http://en.wikipedia.org/wiki/G%C3%B6...teness_theorem
the corresponding Gödel sentence G asserts: “G cannot be proved to be true within the theory T
as was pointed out many years ago if you use or create impredicative statements then what you will get is paradox
and as dean has shown
that is what happens with godels theorems

philosophers such as russell
mathematicians such as poincare
have outlawed these statements from mathematics as they lead to paradox in maths

why
because if godels theorems are true
then they apply to godels theorems as well

if godels first theorem is true then it applies to it self

it is shown by colin leslie dean that Godels first theorem ends in paradox

it is said godel PROVED
"there are mathematical true statements which cant be proven"
in other words
truth does not equate with proof.

if that theorem is true
then his theorem is false

PROOF
for if the theorem is true
then truth does equate with proof- as he has given proof of a true statement
but his theorem says
truth does not equate with proof.
if godels second theorem is true
then it applies to it self
http://en.wikipedia.org/wiki/G%C3%B6...second_theorem

The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

now this theorem ends in self-contradiction

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done
as was pointed out many years ago if you use or create impredicative statements then what you will get is paradox
and as dean has shown
that is what happens with godels theorems

86. No, actually you can't, because circular logic does not allow you to evaluate if a statement is true or false.
That,s right within a system working actively with it. But I mean a cirkle logic between the working system and it,s results and the axioms it holds, the system itself. For instance Mass is regarded as a property of things. Weigth is a result of it. Explaining the latter you will use mass and explaining the first you will use the notion of weight. This circlelogic is closed by the definition of force as N's. But using N's you explain weight also from mass as one N(ewton) is defined using the notion of mass (and gravityforce) Because of this definition of what a Newton is the notion of mass is not proven but based on a circle logic within the system when you include the definition for the units to the system (and why not). Hence mass as property is ontological. That cirkle-logic gives it a certain consistency but Godell sees that as not a proof, the system closes because of the cirkle logic and that makes it impossible to proof or disproof it in relation to reality. Impossible to disproof but also to proof. Impossible to disproof is not a proof.

just to clear up this issue: "Consistency is not provable" is not rephraseable as "there is no provability for consistency or inconsistency".
I know there are systems that can be prooven to be inconsistent. But these can be sorted out at forehand and thrown in the dustbin. After sorting the systems that cannot be prooven consistent in relation to reality (of which they give a certain reflection) remain but by definition also not inconsistent. If you could proove any of these inconsistent they would go to the dustbin also. What is left after sorting out the systems that are obviously inconsisten are the systems where the theorems are about. So no proovability according to the theorem,s, not in one way (consistent) and not in the other way (inconsistent).

Godel,s theorems are not about those systems that are prooven inconsistent (why would he bother about these as they are prooven inconsistent allready) but those that lack proovability in either way...be it consistent (real) or non consistent (false, misleading etc).

So you can go both ways. You could use godell in defence of a system saying that the a system is not proovable anyway (so don,t ask). Or you can argue that a defence of a system lacks any ground, reasoning whatever.

The solution (or opening) in this could be to understand the difference between consistency/inconsistency and completeness/incompleteness. With some incompleteness (simplyfication) a system might work in most cases in giving some pragmatic results. For instance Newtonian fysics works to an extend but never completely accurate. It is consistent in itself but Einstein prooved the incompleteness of it.
But by doing that he did not proof the completeness of his own system(s) and theory though.

I think any system that is consistent in itself would be consistent because it somehow contains a cirkle logic of axioms and the system of a whole.

There is a certain lack of openness that makes it a system. That lack is the closedness that circle logic brings. If reality would be without circle logic then a theory of everything for instance is an impossibility by definition. And as I understand that is a conclusion Godel took from his theorems.

87. Oh yes, now I got what you are saying. I can't be bother to explain edam421 what 'consistent' and 'complete' means so I'll only reply to you.

The example I gave correct only to the extent of showing what a set of inconsistent statements mean. The analogies with physical system do not work, because they are not closed systems. Paradoxes in physical systems arise due to our incomplete knowledge of how they work (i.e. of their inner laws): we basically don't know all the laws (equivalent, in principle, to axioms from mathematics), or we don't know all of them correctly, so what we observe seems paradoxal to what we know. Nonetheless, since it is quite obviously happening, we cannot quite say that the world is quite goddamn clearly inconsistent and obviously wrong (although, out of anger, we usually say so ). It would be a problem if we suddenly knew all the rules of the Universe correctly, and yet some things would happen which would be paradoxical. That would be grand indeed and I'd even be persuaded to pick a religion according to it.

On the other hand, with a closed mathematical system, we already know all the laws (i.e. every axiom) that governs it. In formal logic, 'true' and 'false' are mutually exclusive -- you cannot have something that is both true and false -- so if you end up with such a statement, your system must have gotten it wrong at least as formal knowledge is concerned.

Such a system is not necessarily wrong in terms of it's fundamentally flawed, out of this world and a complete idiocy -- but it's not too useful because it cannot tell you anything definite. Incomplete systems are the best we can do -- it's true they only get us so far, not being able to prove some statements -- but at least they get us to some place as far as knowledge is concerned. A system that can prove something to be both right and wrong isn't too useful as far as science is concerned.

A theorem is not a truth by the way. It can be controversial, intriguing because of the problems it reveals. Taoism for instance you have this kind of playing with paradoxes also. Maybe Godell just woke up one morning having some thoughts and wrote these theorems down ,liked them and decided to publish them. Never meant to take them as "truths" or a set of rules on their own.
Again, it is extremely important to make a distinction between "this" world and whatever mathematical system we're talking about. ZFC can be used to model some aspects of this world -- in other words, there are some things that behave similarily in them -- but that's all, they're entirely different worlds. In terms of absolute cosmic wisdom, a theory is not a fundamental, universal truth (hell, just take the theorem that says the sum of angles in a triangle is 180 deg. and try to test it on a triangle drawn on a sphere). However, in the context of a particular mathematical system, a theorem is a truth.

Edit:

As for the circular logic, in your example this is actually resolved axiomatically. Mass is not defined in terms of gravitation -- it wouldn't make much sense to do so in fact, because two objects of the same mass are subject to different weights depending on what planet they're on, or even depending on what place the Earth they're on. Instead, mass is a fundamental unit adopted by convention, and forces is defined according to it (along with length and time intervals, which are also units adopted by convention).

Extra edit: Circular logic within a system is not usually a sign of inconsistency, it is usually a sign of incompleteness. For instance, if you take Euclidean geometry but exclude the parallel postulate, you end up with a system inside which you get circular logic.

88. Oh yes, now I got what you are saying

as pointed out by colin leslie dean in another thread
the problem with godels theorems is they apply to themselves
thus they are impredicative and has been pointed out by others this leads to paradox
godels first theorem is both applying to itself
and as part of the proof creates an impredicative statement

it is very clear
thus godels second theorem ends in paradox- as it applies to itself
godel is developing a proof with in a system of mathematics
which he takes to give a true theorem

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

89. Please shut up. The theorems do not apply to themselves. They are statements about systems, not theorems. Also, CLD is not registered on this forum (unless you are CLD himself, which explains why you seem to have no clue what you are talking about).

90. Please shut up. The theorems do not apply to themselves. They are statements about systems, not theorems.

and systems created the theorem
you really dont see do you
godels second theorem talks about systems which includes the very systems that created the theorem
thus the theorem applies to itself

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

91. godels second theorem talks about systems which includes the very systems that created the theorem
If you actually read the theorem you will see that it does not apply to the system within which Godel's theorem is created. Even if it did, he would only prove that it is consistent, not that it is also complete, which is required in order to rule it inconsistent as you say.

92. Originally Posted by AlexandruLazar
Extra edit: Circular logic within a system is not usually a sign of inconsistency, it is usually a sign of incompleteness. For instance, if you take Euclidean geometry but exclude the parallel postulate, you end up with a system inside which you get circular logic.
No, you do not. In this case you get a set of axioms that can admit different geometries.

Non-Euclidean geometries were discovered precisely by omitting the parallel postulate and attempting to derive it from the remaining postulates. That effort failed when it was found that there are completely consistent geometries lacking the parallel postulate.

Circular logic refers, not to axiom systems, but rather to invalid logic in which one actually assumes, usually through a subtle mistake, the intended conclusion of a theorem. It is more commonly called circular reasoning.

http://en.wikipedia.org/wiki/Begging_the_question

93. In non-Euclidean geometries, the parallel postulate is not ignored, only re-formulated. I might be wrong on this one, indeed -- but I think that if you take the Euclidean geometry and simply leave out the parallel postulate and try to prove it with the remaining four, you end up with a circular problem (i.e. A is true if B is true, and B is true if A is true). Again, I might be wrong on this one, so if I am, I stand corrected.

94. Quote:
godels second theorem talks about systems which includes the very systems that created the theorem

If you actually read the theorem you will see that it does not apply to the system within which Godel's theorem is created. Even if it did, he would only prove that it is consistent, not that it is also complete, which is required in order to rule it inconsistent as you say.
godel is useing a a mathematical system
his theorem says a system cant be proven consistent
this must then apply to the system he used to create the theorem
thus his theorem applies to itself

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

95. Originally Posted by AlexandruLazar
Look, let's put it this way. Godel's second theorem can be rephrased, by your quote, as follows:

If a system can

A. Be proven to be consistent from within itself AND
B. Be proven to be complete from within itself

then it is inconsistent.

It is your quote, word by word, here it is for reference:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.
CLD is trying to show that logic can be proven to be consistent from within itself, and as per Godel's theorem, it must be inconsistent. However, this is not what Godel's theorem says (which you can see right above: the system must meet both A and B, not just A).

Godel's logic does not meet the criteria B. I'm not even sure if it meets A, but let's assume CLD is right (which is unlikely, but what the heck) and it does. It still does not meet B. There is no paradox here, as CLD implies. He is not applying the theorem correctly. His theorem does not imply that a system cannot be shown to be consistent. As long as it can be shown it is consistent, but not that it is complete (or the other way round), everything is fine.
This is quite confusing, and I think wrong.

We need a couple of definitions.

Provable means provable within the system by formal first-order logic.

An axiom system is consistent if no sentence is both true and false. An equivalent definition is that no false statements are provable.

An axiom system is complete if it is consistent and if given a sentence either the sentence or its negation are provable. An axiom system is incomplete if it is not complete.

Since given a sentence either the sentence or its negation are true, in a complete system all true sentnces are provable. In a consistent system no false sentences are provable.

Here is what Gödel proved:

Gödel's first theorem: Given an axiom system sufficiently rich so as to admit the natural numbers, if the axiom system is consistent, then it is incomplete.

Gödel's second theorem: Given an axiom sufficiently rich so as to admit the natural numbers, then if the axiom system is consistent, consistency is not provable.

Corollary: Given an axiom system that admits the natural numbers, if it can be proved that the system is consistent, then it is inconsistent.

Corollary: Given an axiom system that admits the natural numbers, if it can be proved that the system is complete, then it is inconsistent

96. Originally Posted by edam421
colin leslie dean points out The fundamental problem with Godels theorems
are he creates an imprdeicative statement and the theorems apply to themselves ie are impredicative- thus leading to paradox

ieie this is godels impredicative statement used in his first theorem
http://en.wikipedia.org/wiki/G%C3%B6...teness_theorem
the corresponding Gödel sentence G asserts: “G cannot be proved to be true within the theory T
as was pointed out many years ago if you use or create impredicative statements then what you will get is paradox
and as dean has shown
that is what happens with godels theorems

philosophers such as russell
mathematicians such as poincare
have outlawed these statements from mathematics as they lead to paradox in maths

why
because if godels theorems are true
then they apply to godels theorems as well

if godels first theorem is true then it applies to it self

it is shown by colin leslie dean that Godels first theorem ends in paradox

it is said godel PROVED
"there are mathematical true statements which cant be proven"
in other words
truth does not equate with proof.

if that theorem is true
then his theorem is false

PROOF
for if the theorem is true
then truth does equate with proof- as he has given proof of a true statement
but his theorem says
truth does not equate with proof.
if godels second theorem is true
then it applies to it self
http://en.wikipedia.org/wiki/G%C3%B6...second_theorem

The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

now this theorem ends in self-contradiction

http://gamahucherpress.yellowgum.com...phy/GODEL5.pdf

But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done
as was pointed out many years ago if you use or create impredicative statements then what you will get is paradox
and as dean has shown
that is what happens with godels theorems
Colin Leslie Dean is an idiot who understands essentially nothing of the Gödel theorems.

Let's get things straight.

We need a couple of definitions.

Provable means provable within the system by formal first-order logic.

An axiom system is consistent if no sentence is both true and false. An equivalent definition is that no false statements are provable.

An axiom system is complete if it is consistent and if given a sentence either the sentence or its negation are provable. An axiom system is incomplete if it is not complete.

Since given a sentence either the sentence or its negation are true, in a complete system all true sentnces are provable. In a consistent system no false sentences are provable.

Here is what Gödel proved:

Gödel's first theorem: Given an axiom system sufficiently rich so as to admit the natural numbers, if the axiom system is consistent, then it is incomplete.

Gödel's second theorem: Given an axiom sufficiently rich so as to admit the natural numbers, then if the axiom system is consistent, consistency is not provable.

Corollary: Given an axiom system that admits the natural numbers, if it can be proved that the system is consistent, then it is inconsistent.

Corollary: Given an axiom system that admits the natural numbers, if it can be proved that the system is complete, then it is inconsistent

97. take godels second theorem- ie referes to itself

godel is useing a a mathematical system
his theorem says a system cant be proven consistent
this must then apply to the system he used to create the theorem
thus his theorem applies to itself

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

98. Agreed, I'll try to sum it up. If you try to make something out of CLD's so-called proof, you will se the point he is trying to make is this: the formal logic Godel used to reason about axiom systems is an axiom system in itself. CLD argues that if Godel's theorem is true, then the system he used to prove it is thus shown to be consistent, and therefore it is inconsistent by the first theorem.

(Edit: I meant second theorem here, as corrected by edam421 -- since he already quoted it as such, I am leaving it as it is).

Needless to say this is a complete idiocy as Godel's theorem being true or not does not automatically make his formal logic system consistent, or complete (or, to put it more cleanly, Godel's theorem being true does not prove his framework system to be consistent, or complete for that matter). But you are right -- I should have gathered all the definitions in the same place, otherwise it doesn't make much sense to someone reading this topic.

99. Agreed, I'll try to sum it up. If you try to make something out of CLD's so-called proof, you will se the point he is trying to make is this: the formal logic Godel used to reason about axiom systems is an axiom system in itself. CLD argues that if Godel's theorem is true, then the system he used to prove it is thus shown to be consistent, and therefore it is inconsistent by the first theorem.
wrong
dean is saying godels second theorem is inconsistent
by the terms of the second theorem itself
ie the second theorem is self contradictory

godel is useing a a mathematical system
his [second ]theorem says a system cant be proven consistent
this must then apply to the system he used to create the[second]theorem
thus his[second] theorem applies to itself

if godels second theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his second theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his second theorem is true
then he has proven his logic is consistent within that system
but his second theorem says this cannot be done

100. it can only be proven if his logic is consistent within that system
Godel's logic is either correct or incorrect in that system. Being consistent is a property of the system, not of Godel's reasoning. Again: the conclusion of Godel's theorem can not be applied to any theorem. It is not about theorems. It's like applying Newton's laws of motion to describe flashes of light just because something seems to be moving in that case as well.

101. The word theorem (as AlexandruLazar mentioned) might be not the best word for them. Is it even known if it was Godell who started referring to them as theorems ?

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