>>921279

In any logical, mathematical system that is sophisticated enough to include anything like what we call everyday arithmetic, there are always going to be true statements that cannot be proved within that system. Moreover, if such a logical/mathematical system entails a statement or result that "it IS consistent" (rather, that the system and its rules, used properly, never produce contradictory results), then it is actually, rather, inconsistent.

Since being able to /prove/ that something is true, given such-and-such premises, axioms etc, is valued and held as foundational of mathematical knowledge, and since we are very inclined to rely on arithmetic, Gödel's incompleteness theorems tell us "u can't no everyfin, mane, not ever, no matter how hard u try". There will knowably, provably always be un-proveable (which in philosophy of mathematics might be conflated with un-knowable) gaps in mathematics, which delimits knowledge itself. When Gödel's theorems broke, it is not hyperbole to say that they were the biggest development in logic since the ancient world. The incompleteness theorems specifically also did much to scuttle Hilbert and Whitehead/Russell's respective programs.

So, they were a big deal, and a /definite/ result with direct implications for both epistomology and philosophy of mathematics. That is their significance to philosophy. Also the proof did a clever thing about semantically equating statement-strings with the natural numbers that they represent, or something like that, I honestly don't remember the details. Any time a proof is clever/novel like that, mathematicians like it.

>>921417
>u can't no everyfin, mane, not ever, no matter how hard u try

troof

>>921454
you cant know anything analytically, i believe

>>921279
Self reference is troublesome. This sentence is false.

>Also the proof did a clever thing about semantically equating statement-strings with the natural numbers that they represent,
Primitive recursive arithmetic

Any rationalism fails.

>>922366
Not really, any rationalism that uses arithmetic fails.