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 to “arithmetic is consistent” is true but undecidable from within arithmetic.
It also doesn’t imply that the theory composed of arithmetic plus “arithmetic is consistent” is inconsistent, because this theory is more complicated than arithmetic and does not assert its own consistency.
Of course we think the more complicated theory is true and consistent as well, but adding that would just lead to yet another theory, and so on.
If you try to use mathematical induction to form a theory that includes all such statements, that theory will have an infinite number of postulates and will not be able to be analyzed by a Turing machine.
If you try to use mathematical induction to form a theory that includes all such statements, that theory will have an infinite number of postulates and will not be able to be analyzed by a Turing machine.
This part is not quite accurate. Actually, the commonly used theories of arithmetic (and sets) have infinitely many axioms. The actually problem with your approach above is that the theory still won’t be able to prove its own consistency since any proof can only use finitely many of the axioms. One can of course add an additional axiom and keep going using transfinite induction, but now one will finally run into a theory that a Turing machine can’t analyze.
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 to “arithmetic is consistent” is true but undecidable from within arithmetic.
It also doesn’t imply that the theory composed of arithmetic plus “arithmetic is consistent” is inconsistent, because this theory is more complicated than arithmetic and does not assert its own consistency.
Of course we think the more complicated theory is true and consistent as well, but adding that would just lead to yet another theory, and so on.
If you try to use mathematical induction to form a theory that includes all such statements, that theory will have an infinite number of postulates and will not be able to be analyzed by a Turing machine.
This part is not quite accurate. Actually, the commonly used theories of arithmetic (and sets) have infinitely many axioms. The actually problem with your approach above is that the theory still won’t be able to prove its own consistency since any proof can only use finitely many of the axioms. One can of course add an additional axiom and keep going using transfinite induction, but now one will finally run into a theory that a Turing machine can’t analyze.