18. THE INCOMPLETENESS THEOREM
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As with the First Incompleteness Theorem, the Second Incompleteness Theorem holds for any recursive set of sentences in a first-order language which allow one to code and prove enough facts about arithmetic. The perverse consequence of the Second Incompleteness Theorem is that only an inconsistent set of axioms can prove its own consistency. Truth and definability. A close relative of the Incompleteness Theorem is the assertion that truth in N = (N, S, +, ·, E, 0) is not definable in N. To make sense of this, of course, we need to sort out what “truth” and “definable in N” mean here. “Truth” means what it usually does in first-order logic: all we mean when we say that a sentence σ of LN is true in N is that when σ is true when interpreted as a statement about the natural numbers with the usual operations. That is, σ is true in N exactly when N satisfies σ, i.e. exactly when N |= σ. “Definable in N” we do have to define. . . Definition 18.1. A k-place relation is definable in N if there is a formula ϕ of LN with at most v1, . . . , vk as free variables such that P (n1 , . . . , nk ) ⇐⇒ N |= ϕ[s(v1|n1 ) . . . (vk |nk )] for every assignment s of N. The formula ϕ is said to define P in N. A definition of “function definable in N” could be made in a similar way, of course. Definability is a close relative of representability: Proposition 18.8. Suppose P is a k-place relation which is representable in Th(A). Then P is definable in N. Problem 18.9. Is the converse to Proposition 18.8 true? The question of whether truth in N is definable is then the question of whether the set of G¨ odel codes of sentences of LN true in N, pTh(N)q = { pσq | σ is a sentence of LN and N |= σ } , is definable in N. It isn’t: Theorem 18.10 (Tarski’s Undefinability Theorem). pTh(N)q is not definable in N. The implications. G¨odel’s Incompleteness Theorems have some serious consequences. Since almost all of mathematics can be formalized in first-order logic, the First Incompleteness Theorem implies that there is no effective procedure that will find and prove all theorems. This might be considered as job security for research mathematicians.