One to cross off your list

Suppose an undergraduate wrote this:

The first [incompleteness theorem] says that there are truths of arithmetic that are not provable in a consistent first-order logic that can express arithmetic.

You’d patiently explain that there are four things wrong with this. First, it confuses “logic” with “theory”. Second, as you’d remind the student, the first theorem was originally proved for a higher-order theory, and applies to any theory which is properly axiomatized, whether first-order or not. Third, this statement confuses the conditions for semantic and syntactic versions of the first theorem. If you only assume the theory can express enough arithmetic, then you need to assume soundness to derive incompleteness; if you only assume consistency, as in the standard syntactic version, then you have to assume that the theory can represent enough arithmetic (where this is a matter of proving, rather than merely expressing, enough). And fourth, the statement is fatally ambiguous between (a) there are truths not provable in any consistent theory which can represent enough arithmetic, and (b) for any consistent theory etc. there are truths that that theory can’t prove. Given that folk misinterpretations of Gödel trade on that ambiguity, you drill into your students the importance of clearly avoiding it.

Your student goes on to write:

The standard view is that we cannot prove CON(PA), period. (I use CON(PA) as an abbreviation for the sentence that expresses the consistency of Peano Arithmetic.) … However, all that follows from the Gödel theorems is that we cannot prove CON(PA) with mathematical certainty.

Again, you’d start by patiently reminding the student that there is no such thing as the sentence that expresses the consistency of Peano Arithmetic — and that matters because, as Gödel himself later observed, there are sentences that arguably in some sense express the consistency of PA which are provable in PA. In headline terms, it matters for the Second Theorem which consistency sentence you construct, in a way that it doesn’t matter for the First Theorem which Gödel sentence you construct. But second, and much more importantly, it certainly is not the standard view that we cannot prove CON(PA), period. Any good treatment emphasizes that unprovability in PA is not unprovability period. Third, you’d add that it doesn’t follow from Gödel theorems either that “we cannot prove CON(PA) with mathematical certainty”. What’s wrong, for example, in proving CON(PA) from PA plus the Pi_1 reflection schema for PA? If you are mathematically certain about PA and its implications, why wouldn’t you be equally certain about the result of adding instances of the reflection schema? Arguably you should be: but in any case, nothing follows from Gödel theorems about that issue. And fourth, you might quiz your student about what he makes of the Gentzen proof.

Ah, he says,

We need transfinite inductions along a well-ordered path of length epsilon_0 to prove CON(PA) [in Gentzen’s way]. The issue, then, is this: if human minds know the truth of CON(PA) with mathematical certaintly, is the only method by which we do it the use of infinitely long derivations?

But there seems to be a bad misunderstanding here: you’d remind your student that a proof by transfinite induction is not an transfinitely long derivation: it is just a proof assuming that a certain ordering is well-founded.

Unfortunately, those quotations — and there’s more of the same — are not from a student essay but from a book, one published by MIT Press no less, Jeff Buechner’s Gödel, Putnam, and Functionalism (see p. 8 fn. 8; p. 33; p. 39). The book turned up as I dug through the archeological layers on my desk in my Big Book Clear Out: I was sent it some time ago to review. But with garbles like that in a book one of whose main topics is the implications of Gödel for functionalist mechanism about the mind, I’m not encouraged to read any further. Life being short, I probably won’t.

Added later: The Reviews Editor is twisting my arm in a flattering kind of way. Maybe I will review this after all.

8 thoughts on “One to cross off your list”

  1. Life is indeed short, but dispatching an apparently deeply confused book in a published review is a valuable form of community service (all the more so because of its reasonably reputable publisher). The worse the book, the more valuable the service, in fact.

  2. David Auerbach

    A quibble. I think that no sentence that expresses the consistency of PA is provable in PA. You say that some are. I don't think those sentences do express the consistency of PA. Moreover, said sentences are distinguishable from the real ones in a principled way.

  3. David A: yes indeed, we can quibble about that! — so I've slightly changed my original wording to saying that there is an issue, rather can coming down on one side of it.

  4. The idea that incompleteness shows that humans cannot be machines persists because (a) it seems to make intuitive sense, and (b) the people who think it's wrong don't agree about why it's wrong.

    Also, those who attack the idea often seem to be ideologically committed to some version of machine functionalism or strong-ish AI, rather than being impartial. They often seem keen to grasp any technicality they can use against Lucas or Penrose, and to be interested only in rubbishing their arguments (without any charity of interpretation), rather than trying to work out what the strongest position short of theirs might be and to identify why we cannot go further.

    Putnam is in some ways an exception to this, which adds to the interest of his views (imo) — one of the subjects of this book. (I am almost tempted to read it.)

    This book is also (somewhat) interesting in that it's a defence of functionalism that still manages to be in a muddle about Godel.

    So it's possible that review might be a good opportunity at least to clear up some confusions.

  5. I would enjoy reading more about what you say in your objections to the second statement. Does one of your books mention anything to do with it?

Leave a Comment

Your email address will not be published. Required fields are marked *

Scroll to Top