Guest post: Isaac Solomon
January 2, 2012 2 Comments
From a philosophical point of view, the Axiom of Choice (AC) is often a tough sell. On the one hand, there’s something wrong with vector spaces without bases, or fields without algebraic closures (consequences of AC). On the other hand, non-measurable sets.
The most common example of a non-measurable set, the so-called “Vitali set,” is a tranversal of the quotient space . While is certainly a repulsive creature, there’s no reason to believe, a priori, that all other examples of non-measurable sets are as abstruse. To see that that is indeed the case, we look to the opening line of Solovay’s 1970 paper, “A model of set theory in which every set of reals is Lebesgue measurable”:
We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if the axiom of choice is disallowed.
Ouch. Unfortunately, it gets worse. There is no formula for which ZFC proves, “there is a unique non-measurable subset of reals satisfying .” In other words, while non-measurable sets are definable in certain models of ZFC, the theory does not have the capacity to prove that a particular formula “talks about” a non-measurable set. Thus, it appears that AC furnishes a very large collection of sets that even it cannot get a handle on.
In a recent conversation, Andy and I were wondering if there was some formal way of saying that a proof is constructive. We suggested that a proof be called constructive if every existential statement in its deduction had a (first-order) definable witness. So, for example, the proof of the compactness of a metric space would be constructive if, given an arbitrary sequence in that space, one could explicitly outline an algorithm for producing a convergent subsequence.
Thinking more about this, I suspect our definition may be flawed, in that it shifts responsibility for definability to (the theory of the) model at hand, as opposed to the theory we used to generate it. By that I mean that in certain models of ZFC, in which non-measurable sets are not definable, the proof of their existence is non-constructive, but in other models of ZFC, in which non-measurable sets are definable (without parameters), the same proof would be constructive. To me, that seems to miss the whole point of why we pretend to care about constructability in the first place.
Conversely, if you want to switch your perspective to the theory, things become very arbitrary. Given a model , and a deduction in the theory of that model, do we want to look at provably definable objects in all of , or in some sub-theory? In that light, we’re stuck with philosophy again, wondering which theory is most natural for considering a certain deduction.
For the philosophically-minded, the remedy probably lies in intuitionistic, or constructive, logic. Whereas in classical logic, all statements are assumed to be true or false, in constructive logic, a statement is only deemed true or false if there is a constructive proof witnessing that assertion. Put differently, the law of the excluded middle, and double negation elimination, are not axioms of constructive logic. Thus, a better definition of constructability would be that a deduction is constructible if and only if it is valid in intuitionistic logic.
Luckily, because we are mathematicians and not philosophers, we don’t have to worry ourselves with these insipid considerations. The workaround is never to adopt a more conservative approach to mathematics, but to rephrase paradoxes as counterexamples and leave reality for the physicists.