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Topic: § 454 Equality and the axioms of natural numbers
Replies: 30   Last Post: Mar 23, 2014 2:41 PM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: § 454 Equality and the axioms of natural numb
ers

Posted: Mar 22, 2014 5:20 PM

On Saturday, 22 March 2014 19:57:44 UTC+1, Dan Christensen wrote:
> On Friday, March 21, 2014 2:41:17 AM UTC-4, muec...@rz.fh-augsburg.de wrote:
>

> > On Thursday, 20 March 2014 22:35:55 UTC+1, Dan Christensen wrote:
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> > > The real numbers can also be constructed from Peano's axioms.

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> > That was not the question.
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> > > It should be
> > > possible to prove than no irrational number is equal to a natural number.

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> > That was not the question.
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> You wanted to know if it was possible to prove, starting from Peano's axioms, that pi (or any other irrational number?) is not equal to a natural number.

No, here you can read the original question:
>
> > The question was: If the five truncated Peano axioms, not more and not less, define the natural numbers, as you said, what part of the axioms excluded that pi is a natural number?
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> You are grasping at straws, WM. Again, the Peano axioms are just the 5 essential properties of the natural numbers from which it seems all others can be derived.

Of course everything can be derived by adding suitable definitions (like "0 = 1" or like "the difference between two successors is always the same and is called a unit"). But you said that your five truncated axioms, ***not more and not less***, are sufficient to define the natural numbers. Now defend that assertion or drop it.

Regards, WM