Eight - The Critique of Pure Reason

Summary

In Chapter 5 we looked at a theory about natural numbers. Since all recursive relations are n.e. in Q, and the consistency of Q is provable in elementary number theory, we concluded that Q is undecidable. But Q has only finitely many axioms specific to natural numbers, so there is a conjunction A of these axioms. So if we let P be the logic fixed by the language of Q, then a sentence B of that language is a theorem of Q if and only if A → B is a law of logic according to P. By the completeness theorem for first-order logic we mentioned in Chapter 4, being a law of logic according to P may be read equivalently either as being deducible from logical axioms of P by logical rules of inference of P or as being true in all interpretations of the language of P (which is that of Q). So if being a law of logic according to P were decidable, we could apply an algorithm for it to A → B to decide whether B is a theorem of Q, so Q would be decidable. Hence being a law of logic according to P is undecidable.

Suppose, conversely, there were an algorithm α for being a theorem of Q. The property F_Σ of being a proof in Q in which only logical axioms and rules of P are used is a decidable property; just look at the proof to see which axioms and rules are used. So its gödelization F_A is recursive, as is the predicate L_A(x, y), which holds if and only if x is the gödel number of a proof in Q whose last line is the formula with gödel number y. Let B be any sentence in the language of Q and let b be its gödel number. Then B is a law of logic according to P if and only if (∃x)(F(x) ∧ L(x, b)) is a theorem of Q (where F and L n.e. F_A and L_A), so by applying α to the second we could effectively decide whether B is a law of logic according to P.