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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics
Subject: Re: Simply defining =?utf-8?Q?G=C3=B6del?= Incompleteness and Tarski Undefinability away V24 (Membership algorithm)
Date: Thu, 16 Jul 2020 13:32:57 -0700
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olcott <NoOne@NoWhere.com> writes:
[...]
> I understand that Q defines natural numbers so that Q does not assume
> natural numbers. Q defines them as whatevers that have a successor
> function.

Are you sure about that?  Are you sure that there's no possible model
that satisfies the axioms of Q but is incompatible with the properties
of the natural numbers?

Let's consider a collection of axioms simpler than those of Q:

- A set contains a member "0".
- N + 0 = N.
- If N is a member of the set, then N+1 is a member of the set.
- <handwave>Addition is defined as ...</handwave>

Have I defined the natural numbers?  No.  The natural numbers
satisfy all those axioms, but so do the integers, and so do the
reals, complex numbers, and quaternions.

Are there sets that satisfy Q while violating some of the properties
of the natural numbers?  I don't know.  Do you?  If so, how do
you know?

(I'm probably being imprecise about sets, models, and so on.)

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */