Godel’s moment of bullshit came when he decreed that “infinite number of steps” == “impossible.” From that, he claimed “look, here’s a fact about numbers that numbers themselves can’t prove about themselves. Incredible! Here, smoke some of this.” If he wasn’t so stupid, he would’ve realized the very opposite. That he showed it is valid to talk about proofs that require an infinity of steps, and that there’s some facts about systems which require such proofs. Instead, he chose to sensationalize. Zeno did the same thing. He could’ve proved to the Greeks that we pass through an infinity of points at each footstep. Instead, he told them he proved movement is impossible.
In terms of my three classes of infinities, Godel showed that some proofs require #3 or #2 steps (a quanty bigger than any finite number) but then talked as if those proofs required #1 steps (a quantity bigger than anything, a quantity that truly is impossible to traverse). If there was a statement that required #1 steps to prove and was true, then it truly could be said that some true theorems are impossible to prove. Alas, such a thing, obviously, doesn’t exist.
Ignoring for the time being your business about three classes of infinities, do you have any evidence that Goedel sensationalized in the way you are claiming him to have done? Could he not have just stated the dry mathematical facts of his discovery (“Systems of type such and such cannot do such and such…”), undeniable as they are and non-condemnation-worthy as that would’ve been?
“Sensationalize” is too strong a word. But he liked having proved the impossible possible (or vice versa) and ran with it. And, of course, all of his students loved this claim even more than he did, and talked it up. But I do think, if Godel was in the mood, he could have seen that his theorem could have been stated in a rather different language that would not have been as exciting.
To be specific: Godel could have not used the words “cannot be proved” and instead said “requires an infinity of steps [to prove].” To him, it made little difference, and he chose the phrasing that sounded more stupendous. Unfortunately, he shut the door to later thinkers examining this “requires an infinity of steps” and considering whether an infinity of steps did or did not amount to impossibility. It wasn’t until the study of infinite sets millenia later, that Zeno’s identical twist of words got the unraveling it deserved.
Well, I ask again, can you back up the claim that he stated his theorem as “cannot be proved, in an absolute sense” rather than “cannot be proved, in such and such a system”?
I mean, to make it absolutely clear, the title of his famous paper was “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”. On any but the most uncharitable reading, that seems to have all the requisite qualifications, does it not?
Reword “qualifications” as “qualifiers”, in case it wasn’t clear.
What I want to say is, Goedel was no idiot. He realized that augmenting the kinds of systems he was looking at with the infinitary omega-rule (“From phi(0), phi(1), phi(2), …, conclude Forall n . phi(n)”) would result in complete systems, but he never claimed or suggested otherwise. He wouldn’t even have been philosophically inclined to do so; in fact, he quite explicitly felt that his theorem did not mark such sharp limits on reasoning ability as others have taken it to imply.
I’ll go ahead and admit I have no idea actually what Godel, himself, said of his ideas. Maybe it was all later individuals who restated his theorems in the ways we usually discuss them today. I’d be pretty interested to find out more about this. I apologize for slighting the man, even if, I think, what people (mathematicians, no less) now say he said is what matters.