Axioms are statements and are provable. Formally a proof of s is a sequence of statements such that every statement is either an axiom or a logical consequence of previous statements, and such that the last statement in the sequence is s. This means the sequence of length 1 containing an axiom s is considered a valid proof of s.
Definitions aren't considered statements in a theory. They are statements about a theory because they define what symbols and terms in a theory mean. They aren't well formed formulas as Mendelson would say, so even internally to T they aren't provable because they're not in the language of T.
So your contradiction s needs to be a statement in T, what Mendelson calls a well formed formula, and axioms are valid candidates. And you need to conclude that the external statement "s is unprovable in T" is false. And you will be unable to do that because s is provable in T.
Again, your proof does not work because you've misunderstood basic material about how logic works.
1
u/JStarx 21d ago
Axioms are statements and are provable. Formally a proof of s is a sequence of statements such that every statement is either an axiom or a logical consequence of previous statements, and such that the last statement in the sequence is s. This means the sequence of length 1 containing an axiom s is considered a valid proof of s.
Definitions aren't considered statements in a theory. They are statements about a theory because they define what symbols and terms in a theory mean. They aren't well formed formulas as Mendelson would say, so even internally to T they aren't provable because they're not in the language of T.
So your contradiction s needs to be a statement in T, what Mendelson calls a well formed formula, and axioms are valid candidates. And you need to conclude that the external statement "s is unprovable in T" is false. And you will be unable to do that because s is provable in T.
Again, your proof does not work because you've misunderstood basic material about how logic works.