You can construct any natural number without the axiom of infinity, but you cannot construct the set of natural numbers itself. > It seems like you could defin

You can construct any natural number without the axiom of infinity, but you cannot construct the set of natural numbers itself. > It seems like you could define a new class, call them seits, that include both the set as axiomatically defined as well as these sorts of unbounded objects, and then prove that you can define similar operations as exist for sets on seits, and then go on and live your life. Imagine you’re at a vegan restaurant. Your food arrives, you pull a fistful of ground beef out of your pocket and stick it on top. You’re “still eating vegan food”, you’ve just added beef to it. Finitism is a lot like a dietary restriction. It’s asking “What can you do WITHOUT these axioms?” The question is whether you still have a balanced diet afterwards, whether you’re satisfied, whether you prefer a vegan or omnivorous diet, etc. If you add infinities to your finite sets but call them classes instead you’re just cheating on your diet. (Or you’re talking about models of finitism.) I suspect that there are a ton of results that can’t be proven without the axiom of infinity, but I don’t keep track of them.