## Gödel mangled

Here is E. T. Jaynes writing in Probability Theory: The Logic of Science (CUP, 2003).

A famous theorem of Kurt Gödel (1931) states that no mathematical system can provide a proof of its own consistency. … To understand the above result, the essential point is the principle of elementary logic that a contradiction implies all propositions. Let A be the system of axioms underlying a mathematical theory and T any proposition, or theorem, deducible from them. Now whatever T may assert, the fact that T can be deduced from the axioms cannot prove that there is no contradiction in them, since if there were a contradiction, T could certainly be deduced from them! This is the essence of the Gödel theorem. [pp 45-46, slightly abbreviated]

This is of course complete bollocks, to use a technical term. The Second Theorem has nothing particularly to do with the claim that in classical systems a contradiction implies anything: for a start, the Theorem applies equally to theories built in a relevant logic which lacks ex falso quodlibet.

How can Jaynes have gone so wrong? Suppose we are dealing with a system with classical logic, and Con encodes ‘A is consistent’. Then, to be sure, we might reflect that, even were A to entail Con, that wouldn’t prove that A is consistent, because it could entail Con by being inconsistent. So someone might say — students sometimes do say — “If A entailed its own consistency, we’d still have no special reason to trust it! So Gödel’s proof that A can’t prove its own consistency doesn’t really tell us anything interesting.” But that is thumpingly point missing. The key thing, of course, is that since a system containing elementary arithmetic can’t prove its own consistency, it can’t prove the consistency of any stronger theory either. So we can’t use arithmetical reasoning to prove the consistency e.g. of set theory — thus sabotaging Hilbert’s hope that we could do exactly that sort of thing.

Jaynes’s ensuing remarks show that he hasn’t understood the First Theorem either. He seems to think it is just the ‘platitude’ that the axioms of a [mathematical] system might not provide enough information to decide a given proposition. Sigh.

How does this stuff get published? I was sent the references by a grad student working in probability theory who was suitably puzzled. Apparently Jaynes is well regarded in his neck of the woods …