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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 11:32 AM

On Saturday, 22 March 2014 09:57:53 UTC+1, Virgil wrote:

> > How can the reader determine whether two numbers (or members) are equal or
>
> > not equal. What is the definition of equality used here?
>
>
>
> Identity, of course!

Is identity equal to (numerical) equality?
Is identity identical to (numerical) equality?

>
> Since "x" and "y" and "f(x) and "f(y)" are merely names, x = y if and
>
> only if "x" and "y" are merely different names for the same thing,
>
> and f(x) = f(y) if and only if "f(x)" and "f(y)" are merely different
>
> names for the same thing.

How can you prove that Peano axioms concern numerical equality?
How can you determine numerical equality before numbers are known?
And if not numerical equality, how can you exlude that the axioms give
da, deux, dvo, due, duo, dvi, (dve), ???, to, tva, twa, two, zwei (zwo)?
If you cannot exclude this, how can these axioms, "not more and not less", define the natural numbers?

>
> Thus, for example, one = ein and two = zwei

How can you prove that eins is not zwei? Is zwo the same as zwei? Is due same as twa? What has this all to do with successors?

Regards, WM