I'm curious which axiomatic frameworks Gödel's incompleteness theorems apply. Do the theorems apply to any axiomatic system that adopts potential infinity? Or just to those that adopt completed infinity? Are the "potential infinity only" systems safe from the incompleteness theorems?
Let me be more precise.
If one were to adopt the framework that there is no largest natural number, but does not accept that ALL the natural numbers can be gathered into one unified whole, collection, or "set". Do the incompleteness theorems apply to this?
Godel's Incompleteness Theorems hold in any system that's rich enough to code formulas and satisfaction via natural numbers. You don't need $\omega$ to be a set to do that, so you don't need the Axiom of Infinity. (You also don't need the Power Set Axiom.) --Bob