0 is not equal to 1, so it's not inconsistent. I don't understand what you are trying to say with that.
It would be really silly for a system not to believe it was consistent. And, if it were true, it would also apply to the mathematicians making such statements. The mathematicians are assuming it's true, and it is obviously true, so I don't see why a proof system should not have it.
In any case I don't see how my system requires proving "x is provable implies x". It searches through proofs in a totally unspecified proof system. It then proves the standard halting problem on a copy of itself, and shows that it will never halt. It then returns false, causing a paradox.
Are saying that it's impossible to prove the halting problem?
everything is provable in an inconsistent theory
So if something is not provable in a theory, that proves it is consistent?
I did read your link but I don't understand most of it.
TezlaKoil doesn't include his whole argument here. Basically he is using Gödel's second incompleteness theorem. The theorem proves that a theory sufficiently complex to express arithmetic cannot have a proof of the statement corresponding to "this theory is consistent" without being an inconsistent theory.
This doesn't show that arithmetic has a proof of "this theory is inconsistent" either. If it does, then arithmetic is in fact inconsistent. Since we think arithmetic is consistent, we think that the arithmetical formula corresponding...
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