The title kind of says it all. I've been working through Axiomatic Set Theory, Suppes and Mathematical Logic, Kleene. And I haven't thoroughly studied ordinals and incompleteness yet. But, skimming through it I started thinking about this question.
In suppes, the set of natural numbers, $\omega$ is constructed using the ZFC axioms. However, this seems like it goes against Gödel because ZFC set theory is a consistent theory.
Where is the discrepancy? Is there a limitation in the definition of $\omega$? Suppes later constructs the rational numbers out of the ordinals. Do these rational numbers to satisfy the field axioms?
I suspect what you're thinking is something like:
The issue here is that when we say
we have to be very careful what we mean. Here are a couple of true statements:
No consistent recursively axiomatizable theory extending (in any of a few different senses) PA can prove its own consistency.
If PA proves "PA is consistent," then PA is inconsistent.
If ZFC proves that PA is consistent and PA proves that ZFC is consistent, then each is inconsistent.
The takeaway, essentially, should be "any theory proving the consistency of PA must be significantly stronger$^*$ than PA" - but that doesn't mean that no nice theory can prove the consistency of PA! ZFC proves the consistency of PA, is recursively axiomatizable and stronger (in an appropriate sense) than PA, and - as far as we know currently - is consistent (in particular, ZFC doesn't prove its own consistency).
Put another way, here's the right conclusion: since ZFC proves that PA is consistent, ZFC must be significantly more powerful than PA (and indeed, we know lots of ways in which it's immensely more powerful). In particular, when you write
you're taking for granted the consistency of ZFC itself, which is even less trivial than the consistency of PA.
$^*$Making this fully precise is a bit subtle - ignore it for now.