[Virginia-American]

In other words, there are proofs that require an infinite number of steps?

I don't know. All I said was that the number of steps is unbounded.

I guess it'd help me if I understood whether we are talking about proofs in general or only about algorithmic "proofs."

All proofs in a formal system.

[LibWhacker]

Yeah but, if there are theorems that require more steps to prove than any natural number, then there are theorems that require an infinite number of steps. This follows immediately.

Holy cow, this is really amazing to me, VA. My inclination is to say that any proof of any theorem that never gets to the punch line is no proof at all. Surely such an important theorem has a name. Please tell me what it is so I can go look it up. Thanks. I mean, it's embarrassing; I have an advanced degree in mathematics and I've never heard of it! :-(