Re: Godel's Theorem and Model Theory

On Jul 26, 6:00 pm, Newberry <newberr...@xxxxxxxxx> wrote:
On Jul 26, 1:09 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:

Newberry says...

On Jul 26, 7:08 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
Newberry says...

Assume the standard model
Then T(G)
G [by T(G) <--> G]
this leads to a contradiction
Therefore the standard model is incorrect
That leaves the non-standard models
So let's assume a non-standard model
If I am not mistaken a contradiction can be derived as well
The liar strikes back

What is wrong with the above reasoning?

Well, the key statement that is wrong is the
claim "...this leads to a contradiction". No,
it doesn't. I can't say what's wrong with your
reasoning since you didn't say why you think it
leads to a contradiction.

We have derived G, which says about itself that it is not derivable.

G is constructed for a specific theory. For example, Peano
Arithmetic. For PA we can construct a sentence G such that

G is true (in the standard model of arithmetic)
<-> G is not provable in PA

That's not a contradiction.

When you say "G is true in the standard model" is it not the same as
"if we assume the standard model then G"? .


No! There is no "we assume the standard model". What do you think a
model IS? What we assume or don't assume are sentences or sets of
sentences. A model is not a sentence and not a set of sentences. Of
course, for each model M, there is the set of set sentences that are
true in that model (that is called 'the theory of model M'), but to
say that a sentence S is true in a model M is NOT equivalent (not even
CLOSE to be equivalent) to assuming all the sentences that are members
of the theory of the model M.

How can you be so opinionated about this subject, posting all kinds of
pronouncements on it, when you don't even know its basics?

MoeBlee


.

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