Re: Goedel's Incompleteness Theorem and omega-consistency

george wrote:
Scott wrote:
Z(n) appears to presume that to any natural number n,
a closed term Z(n) can be assigned.

This is not merely a presumption. They in fact do have a
demonstrably constructible specific closed term in mind.
Z(n) is s(s(s(...(s(0)))), with n s's, as Rupert just said.

If [Neg (v Gen a)] belonged to Flg(c) for some class-sign a,
then couldn't we just add a new constant symbol 'x' to the theory as
an instance of such a counterexample, falsifying the first half of
the condition for omega-inconsistency?

I can't identify any "halves" in your quotation of the definition.
What piece specifically is this falsifying?
Write out the whole definition AGAIN and HIGHLIGHT the PART of it
that this is falsifying.

First half: (n) [Sb(a, v, Z(n)) belongs to Flg(c)] -or- all closed terms
satisfy Q(x)

Second half: [Neg (v Gen a)] belongs to Flg(c) -or- [Ex ~Q(x)] is a theorem.

Both conditions must be met for a system to be called omega-inconsistent.

Now add a new constant 'c' to the theory such that ~Q(c). This

1. *Instantiates* the second condition, and
2. *Falsifies* the first.

It is a new closed term that does not satisfy Q(x). From outside the theory
of nonstandard arithmetic, 'c' would be called a supernatural number
constant.


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Relevant Pages

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