Huh, framed like that, that seems like a wild statement considering he later went on to formulate his ontological “proof”, which attempts to prove God’s existence without relying on axioms (and in my not-so-humble opinion fails to do so, because it assumes “good” and “evil” to exist).
But what I’m reading about his incompleteness theorems, it does seem to be a rather specific maths thing, so would’ve been a big leap to then be discouraged in general from trying to do proofs without axioms.
I don’t know much about this, but I can’t help but think that “complete” and “consistent” are doing a lot more work in that sentence than my current understanding of the terms would lead me to believe.
I am sure there was a typo, it’s Gödel’s incompleteness theorem which proves that consistent systems are incomplete.
Consistency means likely what you expect: it’s that you cannot reach contradiction from very axioms.
The result is insane in my opinion, it means any sensible math system with basic arithmetic has a proposition that you cannot prove. AND you cannot also prove that the system is contradiction-free.
It is completionist’s worst nightmare.
The result is insane in my opinion, it means any sensible math system with basic arithmetic has a proposition that you cannot prove.
Stated more precisely, it has true propositions that you cannot prove to be true. Obviously it has false propositions that can’t be proven, too, but that’s not interesting.
Gödel’s incompleteness theorem is the dumbest and least profound theorem ever made.
e.g.- the second sentence is false.
- the first sentence is true.
omg! reality is crumbling!
It was a response to philosophers who were trying to come up with a robust axiomatic system for explaining math. Russell and Whitehead’s Principia Mathematica attempted to formalize everything in math, and Goedel proved it was impossible.
So yes, it’s a bit of a circlejerk, but it was a necessary one to break up another circlejerk.
that seems fine for it’s purpose, but the modern interpretation is dumb. Even the way Godell went about proving it is dumb.
how about:
x = 2
2x = 3,000
omg! they’re inconsistent! big whoop….it just irks me because i saw all these smarty pants talking deep stuff and referencing how the incompleteness theorem changed everything, and it was basically just stoner philosophy, with a rigorous, needlessly convoluted proof.
when i finally read it, being sure it was over my head, it was like cracking a secret code that turned out to say “drink ovaltine”with a rigorous, needlessly convoluted proof.
Again, Goedel’s theorem was in direct response to Russell and Whitehead spending literally decades trying to axiomize mathematics. Russell’s proof that 1+1=2 was 300 pages long. It was non-trivial to disprove the idea that with enough formality and rigor all of mathematics could be defined and proven. Instead of the back and forth that had already taken place (Russell proposes an axiomatic system, critics show an error or incompleteness in it, Russell comes back and adds some more painstaking formality, critics come back and do it again), Goedel came along and smashed the whole thing by definitively proving that there’s nothing Russell can do to revive the major project he had been working on (which had previously hit a major setback when Russell himself proved Russell’s paradox).
how about:
x = 2
2x = 3,000
omg! they’re inconsistent!You didn’t define x, the equals sign, the digit 2, 3, or 0, or the convention that a real constant in front of a variable implies multiplication, or define a number base we’re working in. So that statement proves nothing in itself.
And no matter how many examples of incomplete or contradictory systems you come up with, you haven’t proven that all systems are either incomplete or contradictory. No matter how many times you bring out a new white swan, you haven’t actually proven that all swans are white.
And formal logic and set theory may have seemed like masturbatory discipline with limited practical use, but it also laid the foundation for Alan Turing and what would become computer science, which indisputably turned into useful academic disciplines that changed the world.
you have told me nothing new, and made several incorrect assumptions:…
but hey, i commend your efforts.to be fair, none of us think you understand godel’s incompleteness theorem
to be fair, i don’t think you speak for “all of us”, i also don’t think you’ve actually read it.
I’m not sure who hurt you, but I hope you get better soon.
creative trolling tactic, at least
Incompleteness means that maybe it’s actually impossible to prove or disprove the Riemann hypothesis, or whether P=NP. I think that’s a big deal. Maybe the fact that there are meaningful results shown to be unprovable (like axiom of choice) is more satisfying?
that’s not a big deal. Some things are unprovable… that’s pretty obvious.
also P≠NP, that’s just dumb.
You can never prove it because you don’t know what new math will be discovered…
While what you write is not too far from the conclusion of Gödel’s proof (he proves you can construct a statement which is equivalent to “this statement is not provable”), the point of Gödel’s theorem is that you require a minuscule amount of language (just enough to work with numbers) to do this.
English is very complicated and not a very formal language, so it’s less surprising that you can come up with unprovable statements like yours. Building such a statement in a language that can barely talk about arithmetic is not obvious at all in my opinion. People had already spent a good amount of time choosing a system of axioms that made certain paradoxes impossible to write (for example Russel’s paradox, “does the set of all sets that don’t contain themselves contain itself”, can be written in english, but not in ZFC, the most commonly used axiomatic system in math), and they thought they reached a point where they had fixed all of these paradoxical statements, but Gödel proved not only that they were wrong, but that their goal of a perfect set of axioms where everything could be proven or disproven was impossible to reach.
Also, there are unprovable statements that don’t look anything like yours, like the continuum hypothesis: “there is no infinity that is larger that the number of natural numbers, but smaller that the number of real numbers”. This a perfectly reasonable statement, not only in english but also in ZFC, which looks like something we should be able to either prove or disprove, but in fact we can’t do either. If you want you can add it (or it’s opposite) to the axioms of ZFC, getting a new set of axioms, and you shouldn’t find any inconsistencies. After Gödel’s proof people started asking themselves “is this thing that I’m trying to prove even provable?”, which I don’t think happend very often before.
By the way, this ability to talk about arithmetic is fundamental to the proof: euclidean geometry can’t do that, so Gödel’ theorem doesn’t apply, and it turns out that it’s both consistent and complete.
honestly i don’t find Cantor’s diagonal argument that compelling either. I’ve been working on a counter argument.
In euclidean geometry, you can’t make a circle with the same area of a square… but i guess you can’t express that geometrically? I never really studied it much… but i am fond of geometric proofs.
At any rate, I’ve learned that Wittgenstein provided some great counter arguments to Gödel, so let’s just go with: i may be some heretic, but i’m with Wittgenstein on this.
I don’t expect a lot of open minds on a pro-gödel meme but, i find it annoying how many people act like it’s just soooo profound. I don’t think it is… and i think most of the people claiming Gödel have any understanding of it, and take some interpretation of an interpretation of it and bring it to woo-woo magic land….
(not you, of course, your way of arguing with me is quite refreshing and nice, i wish you were my friend).
At any rate, i wish you well. I guess, Gödel did have his place in that time, i just think people are making a mountain out of a molehill in the present time.
Wittgenstein found an elegant solution to this kind of paradox: “This is obvious bullshit. Next.”
i’m now a fan of Wittgenstein
