Vladimir Voevodsky, one of the world’s best and most influential mathematicians, has stirred up a bit of a hornet’s nest with a video lecture suggesting the possibility that the Peano Axioms — the standard axioms for arithmetic — might be inconsistent.
Since the Peano Axioms are known to be consistent, it’s tempting to dismiss the whole lecture as either a prank or a shocking display of ignorance. The latter temptation is buttressed somewhat by Voevodsky’s bold misstatement of Godel’s Incompleteness Theorem, which plays a central role in the lecture. On the other hand, Voevodsky is smarter than almost anyone else on earth, which earns him the benefit of the doubt — maybe what he’s saying is subtler than it seems. On the other hand, some of those in the “shocking display of ignorance” camp are among the few people in the world who might be as smart as Voevodsky.
To believe that the Peano Axioms are inconsistent, Voevodsky must reject all of the known proofs that they are consistent. In particular, he must reject the simplest and most convincing of all those proofs, which goes like this:
- The Peano Axioms, and therefore all of their logical consequences, are true statements about the natural numbers,
- A collection of true statements cannot contradict itself.
If you were going to reject that argument, you’d pretty much have to reject part a). There are multiple ways to do that. You could deny that there is any such thing as “the natural numbers”, or you could deny that the Peano axioms are true statements about the natural numbers, or you could deny that all of the logical consequences of the Peano axioms are true statements about the natural numbers.
Denying the existence of the natural numbers is pretty much a non-starter. There is a class of people (mostly college sophomores) who refuse to admit the existence of the natural numbers but are nevertheless willing to debate the consistency of Peano Arithmetic (that is, the Peano Axioms and their consequences). But for a mathematician to deny the existence of the natural numbers would be as rare and fruitless as for a psychologist to deny the existence of conscious beings or for a physicist to deny the existence of physical objects. We can’t get by without them.
Besides, it is quite impossible to study Peano Arithmetic unless you know about the natural numbers in advance. For example, a “proof” in Peano arithmetic is a list of statements, each of which is either an axiom or follows from preceding statements. To understand that concept, you must know what a list is. To understand what a list is, you must have the concepts of “first”, “second”, “third”, etc. In other words, you need to know about the natural numbers! Anyone who claims to understand Peano arithmetic has already implicitly admitted that the natural numbers exist — and that you’ve got to be familiar with them before you can axiomatize them.
It would be a bit of an exaggeration, but not much of one, to call this assessment 100% noncontroversial among mathematicians. Almost (but not quite) everyone who studies math, including, I am almost sure, Voevodsky, accepts this account of what we do.
But if Voevodsky, like the rest of us, believes in the natural numbers, then it’s hard to see how he can doubt the consistency of Peano arithmetic. Surely the axioms are obviously true, but Voevodsky seems to be doubting that every logical consequence of the axioms is true. How can this be? If I understand him right (and I’m not sure I do), it’s because he thinks that some logical consequences of the axioms might be so complicated as to be meaningless, and are therefore neither true nor false.
This makes no sense to me at all, because complication does not by itself imply meaninglessness. More to the point, if some consequences of the axioms are meaningless, there must be a least complicated meaningless example. That least complicated example is only slightly more complicated than some other consequence which is meaningful. How can a slight increase in complexity introduce meaninglessness?
Again, I might have missed his point completely. If so, I’m in good company. There have been several threads in the May and June editions of the FOM mailing list (“FOM” stands for “Foundations of Mathematics”) where a mighty distinguished crew have expressed much the same confusion, and, in several cases, dismay.
For those who like this sort of thing, this is the sort of thing they’ll like. You can click on May or June above and select pretty much any thread with the word “Voevodsky” or “consistency” in it. I’m sure there’s content there I’ve failed to digest in full.
Edit: I should have said that in my (possibly flawed) understanding of Voevodsky, it’s not just the consequences of the axioms that can be too complicated to be meaningful, but some of the axioms themselves. For any property that can be expressed in Peano Arithmetic, there is an axiom saying (essentially) that if there are any numbers with that property, then there is a smallest one. V. seems to be saying (and again I want to stress that I’m not sure I understand him) that some of these properties are too complex to be meaningful, and therefore so are the corresponding axioms.