Tuesday, September 6, 2016

The Axiom of Separation

The Axiom of Separation in Zermelo-Fraenkel (ZF) set theory implies that, roughly, for any set A and any unary predicate F(x), there is a subset B of all the x in A such that F(x). But only roughly. Technically the axiom only implies this for predicates definable in the language of set theory. We philosophers tend to forget that technical fact when we use set theory, much as we tend to blithely extend set theory to allow for ur-elements (elements that are not themselves sets). But if we are going to be realists about sets (which I am not saying we should be), we should have a real worry about what predicates can be legitimately used in the Axiom of Separation. (That's one of the lessons of this post.)

Consider the predicate L(x) which holds if and only if someone likes x. This is definitely not formulated in the language of set theory, so ZF set theory gives us no guarantee that there is, say, the set of all real numbers that satisfy L(x). If it turns out that there are only finitely many numbers that are liked, then we have no worries: for any real numbers x1,...,xn, there is a set that contains them and only them (this follows from the Axiom of Pairs plus the Axiom of Union). There will be other special cases where things work out, say when all but finitely many numbers are liked. But in general there is no guarantee from the axioms of ZF that there is a set of all liked numbers.

One might use this to try to get out of some paradoxes of infinity, by limiting the applicability of set theory. That's a strategy worth exploring further, but risky. For the above observations also severely limit the physical applicability of set theory. Suppose, for instance, that at each of infinitely many points in spacetime there is a well-defined temperature. It is usual then to suppose that there is a function T from the spacetime manifold to the real numbers such that T(z)=u if and only if the temperature at z (or, more precisely, at the point of spacetime corresponding to the point z in the mathematical manifold that models spacetime) is u. And we need there to be such a function T to be able to make physical predictions.

One solution is to extend the Axiom of Separation to include some or even all predicates not in the language of set theory. This is the solution that is typically implicitly used by philosophers. The Axiom of Separation has a lot of intuitive force thus extended, but we need to be careful since we know that the incoherent Axiom of Comprehension also had a lot of intuitive force.

A second option would be to have physics make set-theoretic claims. Thus, a theory positing that at each point of spacetime there is a temperature would also posit that there exists a corresponding function from the mathematical manifold that models spacetime to the real numbers. I think this would be quite an interesting option: it would mean that physics actually places constraints on what the universe of set theory is like.

Perhaps if we are not Platonists about sets, things are easier. But I am not sure. Things might just be murkier rather than easier.

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