Thread: How big is Infinity

Originally Posted by river_rat

Originally Posted by epidecus

Ah, I see. And adding more assumptions only opens more doors which leads to more questions. I'm a layman for the realm of set theory axioms, but from I've read, it seems like some hypotheses are unprovable in the standard system of axioms but can possibly be proven in a different system, correct?

Its the proverbial pandora's box actually. We have very little intuition to guide us here, which makes these problems very difficult. It is custom to state any axioms you are assuming beyond ZFC (and sometimes to state if you are using the axiom of choice as well) for any result, which is peculiar if you think about it but that is where mathematics as a disciple is now a days.

You're talking about the formal nature of math, right? If so, I totally agree. From what I've read so far on Wiki, I get the basic idea of the axiom of choice, but I still don't get a few things. If you don't mind explaining in layman terms, why is it treated as an axiom, when to me it seems like just an observed property of sets? And also, why is it so critically important to set theory?

Agreed. Of mathematical interest.

Is this thread going to be infinite?

Originally Posted by question for you

Is this thread going to be infinite?

Only if there is a following post for every post.

Originally Posted by sigurdW

Originally Posted by question for you

Is this thread going to be infinite?

Only if there is a following post for every post.

Good observation sigurd... the logical answer is No. You had a go and thats the main thing my friend

Originally Posted by Strange

Originally Posted by chinglu

If there is no largest natural number, how can you create all natural numbers in any formal reasoning?

You don't have to "create" them, whatever that means. That is like people who say that we don't know the value of pi because you can't write it down in decimal.

Yes, we do have to prove it is creatable. This is a formal theory and does not allow opinions and assertions.

So, it must be decided from the theory that the set of all natural are creatable, that is the nature of the axiom of infinity.

However, I developed a diagonal argument that proves this is impossible.

In terms of ordinal numbers, n = {0,...n-1}.

Now, list all non empty natural numbers by their ordinal representation and in order.

1 = {0}
2 = {0,1}
3 = {0,1,2}

and so on.

Next apply the axiom of choice to each row and select the maximal element in the row.

Then, the range of the choice function satisfies the axiom of infinity since 0 is in the range and n in the range implies n+1 is in the range.

So, the diagonal is the set of all natural numbers.

But, at each choice,the diagonal is also a natural number ordinal So, at choice 1, the range is the natural number 1, at choice n, the diagonal is the natural number n.

In other words, after the nth choice, the diagonal is the set {0.,,,,n-1}, but is also the natural number n. So, if the set on the diagonal emerges into the set of all ordinals, then that set is also equal to the ordinal N, which means N is an element of N which is a contradiction to the axiom of separation.

Originally Posted by epidecus

From what I've read so far on Wiki, I get the basic idea of the axiom of choice, but I still don't get a few things. If you don't mind explaining in layman terms, why is it treated as an axiom, when to me it seems like just an observed property of sets? And also, why is it so critically important to set theory?

It's treated as an axiom because being bale to make arbitrary numbers of finite choices says nothing about being able to make arbitrary number of infinite choices. That would not be so controversial except for the counter-intuitive consequences which can be derived from the axiom of choice.

Originally Posted by river_rat

Originally Posted by epidecus

From what I've read so far on Wiki, I get the basic idea of the axiom of choice, but I still don't get a few things. If you don't mind explaining in layman terms, why is it treated as an axiom, when to me it seems like just an observed property of sets? And also, why is it so critically important to set theory?

It's treated as an axiom because being able to make arbitrary numbers of finite choices says nothing about being able to make arbitrary number of infinite choices. That would not be so controversial except for the counter-intuitive consequences which can be derived from the axiom of choice.

Today we know that any independent axiom can be replaced with a contradicting axiom.
"axiom" no longer mean an obviously true statement. It simply means independent (?) statement in the formulation of a theory.

Originally Posted by sigurdW

Today we know that any independent axiom can be replaced with a contradicting axiom.

True, but kind of by definition as that is what it means for an axiom to be independent (or at least the general proof strategy).

Originally Posted by sigurdW

"axiom" no longer mean an obviously true statement. It simply means independent (?) statement in the formulation of a theory.

Now we are approaching the philosophy of mathematics area...

Originally Posted by river_rat

Originally Posted by sigurdW

Today we know that any independent axiom can be replaced with a contradicting axiom.

True, but kind of by definition as that is what it means for an axiom to be independent (or at least the general proof strategy).

Originally Posted by sigurdW

"axiom" no longer mean an obviously true statement. It simply means independent (?) statement in the formulation of a theory.

Now we are approaching the philosophy of mathematics area...

No accident, Im not a mathematician.

The composition of area as in regards of scaling one point to another being "infinite" does not mean all parts of the area is infinite in spacing, for instance I can hold an infinite amount of different ideas in my brain and therefore the measure of this sort of spacing is technically infinite - but all these ideas may be categorized therefore being sorted and being in their own definition accountable by me. Infinite measured ranged space does not make the area infinite, just in one particular sense.

Originally Posted by Locke23

The composition of area as in regards of scaling one point to another being "infinite" does not mean all parts of the area is infinite in spacing, for instance I can hold an infinite amount of different ideas in my brain and therefore the measure of this sort of spacing is technically infinite - but all these ideas may be categorized therefore being sorted and being in their own definition accountable by me. Infinite measured ranged space does not make the area infinite, just in one particular sense.

Its a TRICK! The brain fools us to believe thoughts to be so! I cant prove that its a conjuring trick...you have to make tests by yourself. All I can do is telling you how I got suspicious and decided that thoughts are finite...(They are real.)

Originally Posted by Locke23

I can hold an infinite amount of different ideas in my brain

Of course you can't. Your brain is finite and therefore can only handle a finite number of thoughts and memories.

Originally Posted by Locke23

The composition of area as in regards of scaling one point to another being "infinite" does not mean all parts of the area is infinite in spacing, for instance I can hold an infinite amount of different ideas in my brain and therefore the measure of this sort of spacing is technically infinite - but all these ideas may be categorized therefore being sorted and being in their own definition accountable by me. Infinite measured ranged space does not make the area infinite, just in one particular sense.

Huh???

Originally Posted by river_rat

Huh???

Oh good. Not just me then.

Originally Posted by Strange

Originally Posted by Locke23

I can hold an infinite amount of different ideas in my brain

Of course you can't. Your brain is finite and therefore can only handle a finite number of thoughts and memories.

Not entirely, as of right now we actually know very little of our brain's greatest potential.

Originally Posted by Strange

Originally Posted by river_rat

Huh???

Oh good. Not just me then.

Well, that hurts my feelings.

Originally Posted by Locke23

Not entirely, as of right now we actually know very little of our brain's greatest potential.

I'm sure most neuroscientists would disagree (even though I am not a neuroscientist). But we do know for certain that it is finite.

Originally Posted by Strange

Originally Posted by Locke23

Not entirely, as of right now we actually know very little of our brain's greatest potential.

I'm sure most neuroscientists would disagree (even though I am not a neuroscientist). But we do know for certain that it is finite.

Really? I was unaware that we had figured that out. I'll have to look it up.

Originally Posted by Locke23

Really? I was unaware that we had figured that out. I'll have to look it up.

Well, if your head was infinitely large, I think astronomers might have noticed by now.

Originally Posted by Strange

Originally Posted by Locke23

Really? I was unaware that we had figured that out. I'll have to look it up.

Well, if your head was infinitely large, I think astronomers might have noticed by now.

Lol. I don't think size is completely the case with storage of data in the brain.

There are still a finite number of cells, a finite number of synapses, even a finite number of atoms and subatomic particles, and a finite number of discrete states all of those can be in. I don't see how that could possibly store an infinite amount of information. Unless you are using "infinite" in the informal sense of "like, loads, I mean, huge amounts".

Originally Posted by Strange

There are still a finite number of cells, a finite number of synapses, even a finite number of atoms and subatomic particles, and a finite number of discrete states all of those can be in. I don't see how that could possibly store an infinite amount of information. Unless you are using "infinite" in the informal sense of "like, loads, I mean, huge amounts".

Anyway, my original purpose of posting here was to say that though some parts of things may be finite, it has other qualities that are of by definition infinite.

Originally Posted by Locke23

Anyway, my original purpose of posting here was to say that though some parts of things may be finite, it has other qualities that are of by definition infinite.

Hmmm... I'm still not sure what that means ...

Originally Posted by Strange

Originally Posted by Locke23

Anyway, my original purpose of posting here was to say that though some parts of things may be finite, it has other qualities that are of by definition infinite.

Hmmm... I'm still not sure what that means ...

Okay then, I tried. My bad.

Originally Posted by Locke23

Originally Posted by Strange

Originally Posted by Locke23

Anyway, my original purpose of posting here was to say that though some parts of things may be finite, it has other qualities that are of by definition infinite.

Hmmm... I'm still not sure what that means ...

Okay then, I tried. My bad.

Try again then. Lets see the definition.