Monday, July 02, 2007

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 ...


Timmo said...

Aren't books usually reviewed by other experts before they are published? From the passage you quoted, it sounds like Jayne's understanding of the Gödel results is really very terrible. It might be a good idea to alert him in a polite letter that his statement of "the essence of the Gödel theorem" is incorrect.

Peter Smith said...

Jaynes's book was published posthumously, so I guess it is just a bit late to send corrections ... politely or otherwise! (At a quick glance, the other logicky stuff at the beginning of the book is pretty bad too.)

a said...

Well just to be argumentative, "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..." is as it stands (at best) misleading. "Elementary arithmetic" could prove its own consistency, and indeed the consistency of set theory; it could prove it because it could itself be inconsistent.

Now of course you may have no doubts that elementary arithmetic is consistent, but that's quite another matter.

By the way you obviously mean "elementary arithmetic" in a technical way. If you mean Q or PA or a system meeting some technical criteria, then you really should say so, because otherwise it seems that you are claiming that any system capable of proving substantial mathematical propositions, is unable to prove its own consistency. And as you know, this is false.

Peter Smith said...

Well, just to be equally argumentative, elementary arithmetic is consistent, and that's a necessary truth, so it couldn't itself be inconsistent!

And "elementary arithmetic" is of course a placeholder in this sort of informal presentation for "enough arithmetic for the usual Gödelian arguments to go through" ... As you know.

a said...

Well, where does the knowledge of the consistency of "enough arithmetic to have the usual Godelian arguments to go through" (let alone its necessary truth) come from? Still I'm not quite sure why its truth or necessary truth doesn't mean it shouldn't be mentioned in your statement beginning with "since" (after all, you're taking Jaynes to task for his lack of understanding of Godel's results; surely precision is in order). Or is the implication that the consistency of "enough arithmetic to have the usual Godelian arguments to go through" is so obvious that it just doesn't need to be mentioned?

And I do certainly prefer "enough arithmetic to have the usual Godelian arguments to go through" to "elementary arithmetic". If that's too long, then use "infinitary arithmetic" or even "weakly infinitary arithmetic" or the like, since the Godelian arguments for the second theorem require the infiniteness of the natural numbers, which is not really so elementary. It would seem to be more informative and less misleading.

Peter Smith said...

Well, of course I agree that -- if on one's best behaviour -- you have to mention consistency assumptions, state how much arithmetic it takes to prove the second theorem etc. But I'm all for the mathematicians' motto "sufficient unto the day is the rigour thereof". It was clear enough, I do hope, what I was originally saying and how to rigorize it. But Jaynes isn't just being a bit unrigorous for the purposes of a quick informal presentation: he doesn't seem to even to be in the right ballpark!