Re: Godel's Theorem and Model Theory
- From: stevendaryl3016@xxxxxxxxx (Daryl McCullough)
- Date: 3 Aug 2007 06:57:01 -0700
george says...
Can #("A") ever produce a hyperfinite natural?
No, if A is a formula of PA, then #A is always
a standard natural. But what's the relevance of
that? My point is that (Ex)(P(x, #("A")) --> A
does not say "There are no hyperfinite numbers".
Come on. It says that ~Con(PA) is false.
Even if it can't make the model standard, it at least narrows it
down to models of PA+Con(PA).
My point is just that "it can't make the model standard".
Your point that it could get the COMPLETE theory of true
arithmetic and STILL allow things nonstandard is one I
still don't appreciate.
The key here is that completeness is with respect to a
particular *language*. If the language is the language
of PA, then let T_true be the theory of true arithmetic,
which is the set of formulas of PA that are true in the
standard model of PA. This theory is necessarily complete,
but it has nonstandard models. To see this, get a new
theory T' with a new constant symbol a and with infinitely many
axioms of the form
a > 0
a > 1
...
(for each numeral n, there is a corresponding axiom
a > n)
This new theory is consistent, but it is not satisfied
by the standard model. So let M be a model of this new
theory. Then since T_true is a subtheory of T', M is
also a model of T_true. So M is a nonstandard model of
the complete theory of arithmetic.
--
Daryl McCullough
Ithaca, NY
.
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