Categoricity

What we said last time about structures suggests that, whatever your attitude about whether mathematics cares about systems of objects only up to isomorphism (that is, only about what is true in and of a structure that is also true in and of any isomorphic structure), it is clearly a central aspect of figuring out what we are talking about to characterize up to isomorphism. Some think that that is not enough (Frege: Julius Caesar =? 17), but everyone agrees that it is central.

People like Frege and Russell thought they knew what the numbers, for example, were up to isomorphism, and so it was the rest they concerned themselves with. However, it has become clear that even figuring out what we are talking about up to isomorphism is nontrivial, and that is our subject today.

Euclid's axioms, notoriously, try to characterize geometry completely, well beyond isomorphism: his first axiom, which usually gets omitted in high-school geometry today (to the extent it still gets taught) states that a point is an object without extent. The notion of extent doesn't ever appear again. The axiom is not there to be used for proving things, it is there to explain what we are talking about based on a prior understanding of extent.

If we eliminate such axioms (and I'll criticize them in a moment), then axioms that hold of a structure hold equally of all isomorphic structures. Thus, the best we can hope for from a system of axioms is a characterization or description of some mathematical system up to isomorphism.

What is wrong with axioms, like Euclid's that attempt to go beyond isomorphism? The axiom accomplishes nothing unless we already understand the notion of extent. It prevents the axioms from being self contained.

If we are using axioms to specify what we are talking about, then what we are talking about can only be characterized up to isomorphism: if the axioms go further, they are relying on notions not given by the axioms to do so, and so, whatever you may say about such axioms, they are not defining, picking out, or characterizing their subject matter.

Axiom systems can have several purposes: one is to characterize their subject matter, another is to serve as a basis for proving things about their subject matter, another is characterizing a subject matter for us, and another is to serve as a basis for us to discover things about their subject matter.

The second pair involves us: it involves the epistemological project of enabling human beings to find things out. For that purpose, axiom systems need to be humanly usable.

First, I want to abstract from epistemological limitations. The axiom system that has the best shot at characterizing, say, the natural numbers, is, given that simplification, immediately obvious: the system of all sentences true of the natural numbers. Since that says as much as possible, it must characterize the natural numbers if any axiom system does.

The shocking fact, discovered by Skolem, is that even that axiom system fails to characterize the natural numbers (up to isomorphism). In fact, following is true: any system of axioms that has any infinite model has an infinite model of every infinite size. This fact is often called SkolemsParadox (Löwenheim-Skolem-Tarski theorem).

So, you may wonder, how do we know what the numbers are? That is a very good question.It seems I must be lying, since Frege and Russell, no idiots, take Peano (really Dedekind) to have given an axiomatic characterization of the natural numbers. So, what's wrong?

We're both right. Everything that I said was clearly imprecise: when I said, all truths about the natural numbers, none of you thought I meant to include that my son is 9 as one of those truths.

I was talking about all truths about the natural numbers expressible using first-order logic in the language of the natural numbers. Frege and Russell had in mind the Dedekind-Peano second order axioms.

-- ProfessorShaughanLavine? - 28 Feb 2005