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The history of axiomatic set theory did not proceed in the way that you suggest here. Zermelo, for example, was motivated to form his axiomatic system primarily in order to give a careful proof of hi …

What references, especially books, have been devoted to specifically addressing the source of the deep roots of the diversity and complexity of number theory? To a first approximation, I would sa …

The short answer is no. Con(T) is a very weak assumption and it is asking a lot for it to have interesting mathematical consequences. A slightly less ambitious question is whether "ZFC + the consist …

There's a general "trick" for handling all issues of this sort. Take any mathematical theorem that a platonist regards as meaningful. Formalize it as a formal theorem T in ZFC. The formalist will n …

One approach, mentioned by Pace Nielsen in the comments, is to start with what I call strict formalism. The only substantive assumption required for strict formalism is that you are capable of recogni …

Harvey Friedman has an example of a finite statement (that's actually "finite" and not "finitary," i.e., $\Pi_0^0$ rather than $\Pi^0_1$) that has a proof, using large cardinal axioms, that is no more …

There's a certain confusion underlying your question, which Andreas Blass's answer is trying to point out. Let me see if I can explain it in different words. You say, “the negation of Con(ZFC) proves …

You have almost answered your own question; it seems that the only part you are confused about is whether "the reflection principle operates outside ZFC." One must, as always, distinguish between the …

The main reference for this topic is Angus Macintyre's appendix to Chapter 1 ("The Impact of Gödel's Incompleteness Theorems on Mathematics") of Kurt Gödel and the Foundations of Mathematics: Horizon …

I believe that what you're looking for is a development of Gödel's theorem in what we might call "the language of syntax" rather than "the language of arithmetic." The closest thing I'm aware of to w …