Atla wrote: ↑Sat Mar 09, 2024 6:43 am
Russell's paradox is invalid because to have a set be its own member is invalid.
Historically, it went exactly the other way around. In order to get rid of Russell's paradox, they decided to add a few axioms to ZFC in order to prevent expressing that kind of paradoxes. They only stopped doing that after Gödel proved that their approach of adding axioms in order to hide problems is essentially futile.
Atla wrote: ↑Sat Mar 09, 2024 6:43 am
If a problem is undecidable, that has no effect on the LEM, we simply can't decide the problem.
How do you even know if P is undecidable? Proving that P is undecidable could be a lot of work and require its own proof. For example, proving that P is independent of ZFC can be a lot of work. We may initially not even be aware of that. For example:
Initially, nobody knew that CH was independent from ZFC. David Hilbert even included the problem in his famous list:
The continuum hypothesis was advanced by Georg Cantor in 1878,[1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900.
David Hilbert even initially assumed that that CH was decidable from ZFC. In fact, he thought that every problem was decidable. With his
Entscheidungsproblem, Hilbert was even looking for proof that every problem was decidable:
https://en.wikipedia.org/wiki/Entscheidungsproblem
In mathematics and computer science, the Entscheidungsproblem (German for 'decision problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm]) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928.[1] The problem asks for an algorithm that considers, as input, a statement and answers "yes" or "no" according to whether the statement is universally valid, i.e., valid in every structure.
The answer to David Hilbert's problem is a resounding "no". There exist fundamentally undecidable problems. That was clearly not the answer that Hilbert was looking for, but hey, it is the only correct answer.
Now, imagine you try to solve a problem P. First, you proclaim "P or not P". Next, you prove that "not P" is false. Next, you therefore conclude that P is true.
Your answer could be incorrect, because P could also be false. Furthermore, as mentioned above, determining the decidability status of a problem can in itself be a lot of hard work. In the meanwhile, you cannot just assume that P would be decidable by liberally proclaiming "P or not P".