Does anyone else find Gödel's incompleteness theorems depressing?

Does anyone else find Gödel's incompleteness theorems depressing?

what is it ?

you can't know nothing

Basically that there is no way to define arithmetic (and thus almost all other forms of mathematics) through a series of axioms (e.g. Peano arithmetic) and be sure that the system is consistent.

I honestly think you can apply his theorem to every realm of human understanding. Paradoxes are a part of our very existence. We probably live in a fractal universe. Everyday we make assumptions about the nature of our lives that are far less rigorous than mathematical ones. The incompleteness theorems combined with the idea of chaos theory are enough to convince me that we will be ignorant for eternity.

Yes, but mathematics and logic, especially when considering we're here constructing the rules and systems from scratch, would seem to be the one area where we can have absolute knowledge and control. But apparently not so. Ergo, depressing. Imagine, no consistent system of mathematics built from simple rules.

Thankfully Godel proved the completeness of first order logic. At least we have that.

no, to me it's keeping things spicy... and it stopped Hilbert in his "everything must be safe"-madness

In my opinion, Gödel only proved that you cannot guarantee absolute truth, not even in self-constructed systems using axioms. And that feels pretty right to me.

The Incompleteness Theorem is not depressing.

I mean, even if you COULD prove that ZFC is consistent within ZFC, what help would that be?

It doesn't imply that the system is consistent, because inconsistent systems can prove everything, including their own consistency.

The question of "is ZFC consistent" will forever remain a mystery, simply because it's the bare basics of mathematics. We can prove the consistency of basically every other system from it, though (by constructing them using set theory, thus showing an example of a model).

The best we could do is try and search for contradictions on ZFC using computer programs (and prove that ZFC has no contradictions which are less than, say, 200 lines long).

What would be depressing would be their negation.

Imagine if there were a consistent complete foundation for mathematics. Then the truth value of literally any mathematical proposition could be evaluated by a computer.

How sad would that be.

I'm not sure I follow.

Which part? That there would exist a computer program that could evaluate the truth value of any mathematical proposition?

If there existed a recursive complete consistent foundation for mathematics, the existence of such a program would follow from Gödel's completeness theorem -- the program would just simultaneously search for a proof of the statement and for a proof of its negation, and would find one or the other, and thus recursively determine the truth value.

Every mathematician would be out of a job, but that's not why it would be sad. It would be sad because mathematics would be a "solved" field. The computer program would be the solution to all of mathematics, ever. Mathematical question? Input it into the computer program. Solved.

It's depressing because he basically defined modern mathematics and logic when he was just 25. Why aren't I that smart?

You're sane.

No - quite the opposite! If any system of information will inevitably result in a paradox that is irresolvable from within said system, then that means there must be a system in which it is resolvable. For example, LSL (the language of sentential logic) has paradoxes that only LMPL (language of monadic predicate logic) can resolve - and in it, things only resolvable in first-order logic, and then second, and so on. Essentially, what this suggests is an Aleph-Null of all known sets of systems of information, which will produce certain paradoxes which can be resolved by stepping outside of any form of symbolization whatsoever, and communicating concepts through pure information itself. It means that any one person's life can be understood as a closed system that might be understood through another. It means that no one source can ever have all the answers, until there is only one source that has all known answers, and that it will be able to communicate understanding without any symbolization or abstraction. It provides a logical framework for omniscience by instantiating a limit for the transmission of information itself.

In what program do you actually learn about this stuff?

Graduate math classes in metamathematics.

No cause I'm not retarded.

Really? My logic course had a part were we constructed his proof.

Had it in a few classes. All of these were undergraduate math courses, except set theory and computability theory also counted as a grad course.

-Intermediate Logic Class (sketched proof)
-Set theory
-Computability theory

drop acid and read GEB

>logical systems are created by nonlogical beings

pretty simple desu senpai

>Every mathematician would be out of a job
That's not really true, a mathematician does more than just verify the truth of statements in some theory, they have to come up with meaningful definitions in order to advance a field. The classic example is topology, the definition of a topology is far from obvious. Even if a computer could, given enough time, answer every mathematical question, it wouldn't be able to ASK the right questions.

This bruh. You'll never be the same again.

Literally the only thing that allows math to be a respected profession is the promise that one day our theorems might be useful for some physicist or engineer. If you can generate theorems from a computer than there is no reason to keep mathematicians around any longer.

Read it from Introduction to Mathematical Logic book. Gödel's theorems themselves are actually pretty easy to understand and prove. It's getting there that's hard.

The affirmative power of falsity will save you my brother.

Both the first and second incompleteness, in an undergraduate logic course? In my undergrad logic course it was more a sketch of the proof, and certainly lacked the generality to showing the incompleteness of other theories; I didn't feel like I truly understood it until it was done more completely in a grad class. But I'm glad your class did that.