Notices
Results 1 to 11 of 11

Thread: The definition of Addition, Multiplication, Exponentiation

  1. #1 The definition of Addition, Multiplication, Exponentiation 
    Guest
    We can define the addition of two numbers, a and b, as such:









    We can define the multipication of two numbers, a and b, as such:





    We can define the exponentiation of two numbers, a and b, as such:





    It is clear to me (and hopefully you) by now why the Dedekind-Peano axioms use a successor, n + 1, axiom to prove arithmetic (this is as basic a definition as we can use). (Of course, I'm aware that there are more axioms, as obviously the natural numbers would have to exist amongst others, the successor axiom is merely the one I am referring to).


    Reply With Quote  
     

  2.  
     

  3. #2  
    Forum Freshman
    Join Date
    Nov 2011
    Location
    Sweden
    Posts
    39
    Quote Originally Posted by ;252178
    We can define the addition of two numbers, a and b, as such:









    We can define the multipication of two numbers, a and b, as such:





    We can define the exponentiation of two numbers, a and b, as such:





    It is clear to me (and hopefully you) by now why the Dedekind-Peano axioms use a successor, n + 1, axiom to prove arithmetic (this is as basic a definition as we can use). (Of course, I'm aware that there are more axioms, as obviously the natural numbers would have to exist amongst others, the successor axiom is merely the one I am referring to).
    The Peano-Dedekind axioms define the successor relation to be a function.

    Using this we can generate all the natural numbers starting from 0.

    They also define mathematical induction, which is a powerful technique of proof and construction to simplify and shorten
    proofs and constructions by other methods (such as iteration).

    Successor and mathematical induction are used to define the operators: addition, multiplication and exponentiation.

    The definitions you are using are basically iterative.


    Reply With Quote  
     

  4. #3  
    Suspended
    Join Date
    Dec 2010
    Posts
    661
    Quote Originally Posted by Elterish View Post
    Quote Originally Posted by ;252178
    We can define the addition of two numbers, a and b, as such:









    We can define the multipication of two numbers, a and b, as such:





    We can define the exponentiation of two numbers, a and b, as such:





    It is clear to me (and hopefully you) by now why the Dedekind-Peano axioms use a successor, n + 1, axiom to prove arithmetic (this is as basic a definition as we can use). (Of course, I'm aware that there are more axioms, as obviously the natural numbers would have to exist amongst others, the successor axiom is merely the one I am referring to).
    The Peano-Dedekind axioms define the successor relation to be a function.

    Using this we can generate all the natural numbers starting from 0.

    They also define mathematical induction, which is a powerful technique of proof and construction to simplify and shorten
    proofs and constructions by other methods (such as iteration).

    Successor and mathematical induction are used to define the operators: addition, multiplication and exponentiation.

    The definitions you are using are basically iterative.
    What?

    The successor function derives its nature from the axiom of infinity.

    Once the successor function is defined in reference to the axiom of infinity, then one can use the ordinality of the natural numbers to prove mathematical induction.

    This is the method in the foundations of set theory.
    Reply With Quote  
     

  5. #4  
    Forum Freshman
    Join Date
    Nov 2011
    Location
    Sweden
    Posts
    39
    Quote Originally Posted by chinglu View Post
    What?

    The successor function derives its nature from the axiom of infinity.

    Once the successor function is defined in reference to the axiom of infinity, then one can use the ordinality of the natural numbers to prove mathematical induction.

    This is the method in the foundations of set theory.
    Yes I'm well aware of axiomatic set theory (I did my masters thesis on it).

    I use Peano-Dedekind theory as its more understandable (for non-mathematicians).
    Reply With Quote  
     

  6. #5  
    Suspended
    Join Date
    Dec 2010
    Posts
    661
    Quote Originally Posted by Elterish View Post
    Quote Originally Posted by chinglu View Post
    What?

    The successor function derives its nature from the axiom of infinity.

    Once the successor function is defined in reference to the axiom of infinity, then one can use the ordinality of the natural numbers to prove mathematical induction.

    This is the method in the foundations of set theory.
    Yes I'm well aware of axiomatic set theory (I did my masters thesis on it).

    I use Peano-Dedekind theory as its more understandable (for non-mathematicians).
    Yes, well, you cannot use Peano-Dedekind theory for set theory based on the 2 incompleteness theorems of Godel unless you can prove the Diagonal Lemma is flawed. More specifically, according to Cantor's theorem, in contrast to his so called diagonal theorem, the power set under ZFC of the natural numbers produces a cardinality greater than that of the model N that satisifies Peano-Dedekind.
    Reply With Quote  
     

  7. #6  
    Forum Freshman
    Join Date
    Nov 2011
    Location
    Sweden
    Posts
    39
    Yes, every theory has a limited domain of validity, This applies both to mathematical and scientific theories.

    We can prove that 2 + 2 = 4 in the natural numbers using Peano-Dedekind theory.

    But to prove, for example, the theorems of analysis (such as the intermediate value theorem) requires the might of axiomatic set theory.
    Reply With Quote  
     

  8. #7  
    Suspended
    Join Date
    Dec 2010
    Posts
    661
    Quote Originally Posted by Elterish View Post
    Yes, every theory has a limited domain of validity, This applies both to mathematical and scientific theories.

    We can prove that 2 + 2 = 4 in the natural numbers using Peano-Dedekind theory.

    But to prove, for example, the theorems of analysis (such as the intermediate value theorem) requires the might of axiomatic set theory.
    You are wrong. There are incomplete theories.
    Reply With Quote  
     

  9. #8  
    Brassica oleracea Strange's Avatar
    Join Date
    Oct 2011
    Location
    喫茶店
    Posts
    16,540
    Quote Originally Posted by chinglu View Post
    Quote Originally Posted by Elterish View Post
    Yes, every theory has a limited domain of validity,
    You are wrong. There are incomplete theories.
    Yes. That is what he said. Which part of "limited domain of validity" did you not understand?
    Reply With Quote  
     

  10. #9  
    Moderator Moderator
    Join Date
    Jun 2005
    Posts
    1,607
    Quote Originally Posted by chinglu View Post
    Yes, well, you cannot use Peano-Dedekind theory for set theory based on the 2 incompleteness theorems of Godel unless you can prove the Diagonal Lemma is flawed. More specifically, according to Cantor's theorem, in contrast to his so called diagonal theorem, the power set under ZFC of the natural numbers produces a cardinality greater than that of the model N that satisifies Peano-Dedekind.
    I warn casual readers that almost all of this is complete gibberish.

    1. The Peano axioms, sometimes referred to as the Peano-Dedekind axioms, are just that - axioms. Things that seem reasonable to all reasonable Men. There exists no "theory" called Peano-Dedekind;

    2. The claim that "you cannot use the Peano-Dedekind axioms for set theory unless the diagonal lemma is flawed" is totally without meaning. There exists NO "diagonal lemma";

    3. The proof that the cardinality of the powerset on the natural numbers is strictly greater than that of is just an application of the diagonal argument of Cantor.

    4. Using this fact, it is possible to demonstrate (albeit rather non-rigorously) the first incompleteness Theorem of Goedel. So this post is sort of on it's head. And that from the world's greatest logician!

    As usual, chinglu appears to post content he fails to understand. Worse, he fails to understand that some members here do understand it
    Reply With Quote  
     

  11. #10  
    Suspended
    Join Date
    Dec 2010
    Posts
    661
    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by chinglu View Post
    Yes, well, you cannot use Peano-Dedekind theory for set theory based on the 2 incompleteness theorems of Godel unless you can prove the Diagonal Lemma is flawed. More specifically, according to Cantor's theorem, in contrast to his so called diagonal theorem, the power set under ZFC of the natural numbers produces a cardinality greater than that of the model N that satisifies Peano-Dedekind.
    I warn casual readers that almost all of this is complete gibberish.

    1. The Peano axioms, sometimes referred to as the Peano-Dedekind axioms, are just that - axioms. Things that seem reasonable to all reasonable Men. There exists no "theory" called Peano-Dedekind;

    2. The claim that "you cannot use the Peano-Dedekind axioms for set theory unless the diagonal lemma is flawed" is totally without meaning. There exists NO "diagonal lemma";

    3. The proof that the cardinality of the powerset on the natural numbers is strictly greater than that of is just an application of the diagonal argument of Cantor.

    4. Using this fact, it is possible to demonstrate (albeit rather non-rigorously) the first incompleteness Theorem of Goedel. So this post is sort of on it's head. And that from the world's greatest logician!

    As usual, chinglu appears to post content he fails to understand. Worse, he fails to understand that some members here do understand it

    I want the casual user to understand this mod has no idea what he is talking about.

    I personally took this course in grad school at UW madison.


    Model Theory CC Chang, H J Keisler, North Holland 1994

    Page 42.

    Number theory (or Peano Arithmetic) has the following list of axioms....

    Page 36

    A (first-order) theory T of L is a collection of sentences of L.

    Page 37

    A set of axioms of a theory T is a set of sentences with the same consequences as T.


    2) The mod is confused about Godel's incompleteness theorems which demonstate using the diagonal lemma that set theory cannot prove its own consistency.

    Here is my statement.
    Yes, well, you cannot use Peano-Dedekind theory for set theory based on the 2 incompleteness theorems of Godel unless you can prove the Diagonal Lemma is flawed.
    Here is the mod's
    The claim that "you cannot use the Peano-Dedekind axioms for set theory unless the diagonal lemma is flawed" is totally without meaning. There exists NO "diagonal lemma";

    Here is the diagonal lemma.
    Diagonal lemma - Wikipedia, the free encyclopedia

    Next, if you understand it, Peano arithmetic cannot replace or be used for set theory just as I said.

    So, as we can see there is a diagonal lemma.
    3) is in another thread

    4) All the statements made by the mod are false so the statement in 4 is vacuous implication.
    Reply With Quote  
     

  12. #11  
    Forum Freshman
    Join Date
    Dec 2010
    Posts
    78
    I can't figure out how to delete my post.
    Last edited by Ledger; November 23rd, 2011 at 03:57 PM.
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •