Re: Successor Axiom: on what grounds TF?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/07/05 

Date: Mon, 07 Feb 2005 05:24:34 -0600

On 6 Feb 2005 06:54:26 -0800, Helene.Boucher@wanadoo.fr wrote:

>
>David C. Ullrich wrote:
>> On 5 Feb 2005 08:03:37 -0800, Helene.Boucher@wanadoo.fr wrote:> >
>> >The idea is not to derive interesting math given a replacement axiom
>> >for the Successor Axiom. It is to see what interesting math can be
>> >derived without assuming the Successor Axiom or its alternative.
>>
>> Strikes most of us as an extremely silly idea. It's like you're
>> saying you want to see what you can derive about the natural
>> numbers without using the fact that they're the natural numbers.
>>
>>
>
>I don't know why you think this is "like". I think Q is more like
>what you're suggesting, because you don't have access to induction.
>And a lot of people like to look at Q.

Not at all. Q is an actual structure, not a list of axioms.
And it's a structure that people are interested in for
natural reasons - what is the intended model for PA minus
the successor axiom, and what does that model have to do
with anything else in mathematics?

>Here are things you can prove in such a system:
>(1) (x)(y)(x + y = z => y + x = z)

Seems a little implausible that one can _define_
x + y without successors. What's the definition
of x + 1?

>(2) (x)(y)(x * y = z => y * x = z)
>(3) etc., i.e. all familiar algebraic laws
>(4) FTOA
>(5) Quadratic reciprocity
>
>It would also seem you could derive Fermat's Last Theorem. Here's an
>extremely brief handwaving argument: FLT asserts that there don't
>exist natural numbers where a certain condition obtains. The SA is an
>axiom which provides, in a sense, *more* natural numbers, so it makes
>FLT more difficult to obtain.
>
>On a technical level it seems interesting that one is still able to
>prove a large class of theorems without the Successor Axiom. If you
>find that "silly", well, that's okay, different tastes for different
>people.

************************

David C. Ullrich

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