Axiomatizable Theories

For the last few lectures, we've been talking about using axioms to characterize mathematical systems, chiefly the natural numbers. However, most of the time, we use axioms for a different purpose: for answering questions, discovering truths, about a system.

For the purposes of characterization, we were maximalist: we allowed any system of axioms, even the set of all truths about some structure as a system of axioms in an attempt to characterize it.

That won't do for the new project: We obviously can characterize all the truths about the numbers starting from a system that includes all the truths. Every theorem has a one-line proof.

That would be tremendously convenient, it would certainly simply proving things, but for one glitch: we have no way of telling what the axioms are, and so no way of using them.

So, for today, a theory is any set of sentences (of first-order logic) that is closed under logical consequence. A set of sentences $S$ is a set of axioms for a theory $T$ if $S\subseteq T$ and if $T$ is the set of all logical consequences of $S$.

We do not work with theories, we work with sets of axioms. Indeed 99% of a mathematician's job is finding out what is in a theory given a set of axioms. For this to be worthwhile, we need to have some way of knowing "in advance" which sentences are in the set of axioms. As a minimum requirement, a set of axioms must be decidable to be humanly usable, that is, there must be a procedure (computer program) that, given a sequence of symbols as input, prints out yes, if the sequence is an axiom and no if it is not.

The requirement of decidability looks much too weak: surely there are sets of axioms that are decidable in this technical sense that are still too awful to be usable. However, the usable systems are certainly decidable, and so negative results about decidable systems of axioms will apply to humanly usable systems.

Moreover, Vaught proved the rather surprising result that if a theory is sufficiently rich (allows Gödel numbering) and it has a decidable system of axioms, then it is axiomatizable by a single schema.

We say that a theory is axiomatizable if there is a decidable set of axioms for it.

Gödel showed that the theory of arithmetic (that is, all truths about the numbers) is not axiomatizable. That poses a problem for formalists, since no formal theory (at least if a formal theory must be axiomatizable) captures what is true about the natural numbers, and therefore there is no formal theory the study of which is a good substitute for the study of the natural numbers.

Moreover, Gödel's proof does far more: It shows us how, given a decidable axiomatization that is true of the natural numbers, it is possible to find a sentence true of the natural numbers that is not a consequence of the axiomatization. Thus, we are in a position to know more about the natural numbers than follows from any axiom system we know about.

The "essential undecidability" of arithmetic suggests an obvious strategy. Start with, say, PA, construct the Gödel sentence, say, G, move to PA+G, contruct the Gödel sentence, add that on, ... . Those three dots connote a procedure iterated over a recursive ordinal. Here is the problem: there are many ways of setting up notations for recursive ordinals. None of them captures all the recursive ordinals. That is, a given notation peters out after a while. There is always a way to switch to a more sophisticated notation that goes further. There is no single way of systematically going all the way up. (The name for the mathematical study of this kind of idea is inexhaustibility).

At any rate, we can't use axioms to characterize the numbers, and we can't use axiomatizable theories to discover all the (first-order) facts about the numbers.

The situation is Euclid's nightmare. Since the only satisfying picture we ever thought we had or might have had about mathematics was Euclidean, we are in trouble.

-- ProfessorShaughanLavine? - 04 Mar 2005