Godel's Theorem and Model Theory

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

.

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