30 - Undecidability and incompleteness

Summary

Theorem 13.6 tells us that Q can capture all p.r. functions. Our next theorem shows that Q can in fact capture all µ-recursive functions. With a bit of help from Church's Thesis, our new stronger theorem enables us very quickly to prove two new Big Results: first, any nice theory is undecidable; and second, theoremhood in first-order logic is undecidable too.

The old Theorem 13.6 is, of course, the key result which underlies incompleteness theorems like Theorem 17.2 (if T is nice and ω-consistent, then T is incomplete). Our new theorem correspondingly underlies some easy variations on that earlier incompleteness theorem and its relatives. We'll also prove a formal counterpart to the informal theorem of Chapter 6.

Q is recursively adequate

Recall that we said that a theory is p.r. adequate if it captures each p.r. function as a function (Section 12.4). Let's likewise say that

A theory is recursively adequate iff it captures each µ-recursive function as a function.

We showed that Q is p.r. adequate in Chapter 13. Overall that took some ingenuity; but given the work we've already done, it is now very easy to go on to establish

Theorem 30.1 Q is recursively adequate.

Proof Theorem 13.3 tells us that Q can capture any Σ₁ function as a function. To establish that Q is recursively adequate, it therefore suffices to show that recursive functions are Σ₁ (i.e. are expressible by Σ₁ wffs).