Re: Godel's Theorem and Model Theory

On Jul 25, 4:21 pm, poopdevi...@xxxxxxxxx wrote:
Hi Everybody,

I've been thinking about non-standard models of PA lately, especially
with regards to Godel's Incompleteness theorem. To be clear, here is
the version I'm working from: For each recursively enumerable theory
containing PA, "an arithmetical statement that is true, but not
provable in the theory, can be constructed." -- the quoted part is
from wikipedia.

What exactly is meant by 'true' in the above? The completeness
theorem says that a sentence in a theory T can be proved iff it is
true in every model of T. So if PA can't prove this mysterious
sentence (I know, Godel demonstrates one), it must be false in some
model of PA. Does 'true' mean merely "true in some model"? Or "true
in the standard model"? Something completely different?

For my own sanity, I've always re-interpreted Godel's theorem using
the Soundness and Completeness theorems as essentially stating that
for each recursively enumerable theory T containing PA, there exists a
sentence P true in some model of T and false in others. I'm afraid I
might have been mistaken. However, if I wasn't, is there any way to
characterize the class of P's? Put another way, what would a non-
standard (in this sense) model of PA look like? Countable ordinals?
Cars and boats thrown in?

Thanks,
Alex

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?

For one, Peano attempted to codify the standard model because that is
what we are interested in - that is arithmetic. But if we assume the
standard model we get a contradiction. Is arithmetic inconsistent?

.

Relevant Pages

  • Re: Godels Theorem and Model Theory
    ... That leaves the non-standard models ... If I am not mistaken a contradiction can be derived as well ... claim "...this leads to a contradiction". ... When you say "G is true in the standard model" is it not the same as ...
    (sci.logic)
  • Re: Godels Theorem and Model Theory
    ... Therefore the standard model is incorrect ... That leaves the non-standard models ... If I am not mistaken a contradiction can be derived as well ... claim "...this leads to a contradiction". ...
    (sci.logic)
  • Re: Godels Theorem and Model Theory
    ... That leaves the non-standard models ... If I am not mistaken a contradiction can be derived as well ... claim "...this leads to a contradiction". ... G is true (in the standard model of arithmetic) ...
    (sci.logic)
  • Re: Godels Theorem and Model Theory
    ... That leaves the non-standard models ... If I am not mistaken a contradiction can be derived as well ... claim "...this leads to a contradiction". ... G is true (in the standard model of arithmetic) ...
    (sci.logic)
  • Re: Godels Theorem and Model Theory
    ... containing PA, "an arithmetical statement that is true, but not ... For each consistent recursively enumerable theory containing PA this ... True in the standard model. ... The usual formulation of the theorem is "Given any ...
    (sci.logic)