>>7851523 You can't prove that something is unprovable in general. What you can sometimes prove is that something is unprovable *in a specific system of formal proof*.
Most practical mathematical proofs are not tied to a formal proof system; a proof is a line of reasoning that is judged by other mathematicians to be valid or invalid. While there certainly are clear rules for what makes a proof valid in this way, they are not so clear that a computer could follow them.
What you can do, however, is to make a full mathematical description *of a system of proof*, describing in a completely formal way what constitutes a proof in this system; that is to say, describe it in such a way that you can write a computer program that looks at a candidate proof, and then says "this is valid" or "this is not valid".
If you have defined such a system, you can then prove *in informal mathematics* that no valid proof for some property P exists *in your formal proof system*.
If you can prove this way that no proof for P exists in your system, and you can also prove that no proof for not-P exists in your system, then this shows that your proof system is incomplete; there are things that are true, for which no proof exists in the system. This is of course an undesirable property for your proof system.
I think that the state of being you call unprovable can only exist temporarily. Take for instance near death experiences. There's actually a lot of reports of these, they tend to follow a similar model most of the time, but they truly exist outside the realm of science at this present time. All we have are anecdotal reports. There's at least something chemically in the brain happening, but what that is we can't know, as such experiences happen rarely and very spontaneously.. many people near death or who died temporarilly have no recollection of anything happening whatsoever. And since the bloodbrain barrier and the delicate nature of that particular organ make investigation into the phenomenon a wee bit difficult, it's safe to say this one is a little outside our present pay grade as a species. And that's just looking at it chemically, not extending to any theories of soul or the existential realities present in the confirmation or the denial of such a thing. I mean all the biggest questions in life lie in that improvable wasteland.. science will one day be able to reach them, perhaps, so long as we keep it trucking along. But once we reach them, once we figure out a method of proving there is no way to square a circle, for instance, that whole unprovable thing just gets relabeled.
We really should develop a new language, as a species, which integrates scientific thinking, because this unprovable thing is a linguistic artifact
>>7851623 You are so wrong I cringed. Don't propagate misinformation about shit you don't understand.
a) the only proof systems considered in such contexts are *sound and complete* proof systems. Key word complete. Any statement that is a logical consequence of the axioms is provable in the formal proof system. Thus, statements neither provable nor disprove me truly are logically independent.
b) Proofs of the incompleteness of a given axiomatic system can be formalized within the self-same complete proof system proved to be such that there are statements neither provable nor dis provable in that system.
There truly are statements wholly logically independent of any recursively enumerable axiomatic foundation of mathematics. This is not a consequence of the independence of a given proof system (indeed there are sound and complete proof systems) but of boba-fide logical independence.
>>7851706 right but that's what I'm saying, if you take the set theoretic axioms on their own without the AOC and try to prove the AOC (ie: show that AOC follows from the basic axioms) it has been shown to be both impossible to prove and disprove. Zorn's Lemma as well (which is just equivalent to the AOC, iirc)
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