Comments on: Constructive stone: finite sets

By: Andrej Bauer

Andrej Bauer — Fri, 11 Sep 2009 10:22:06 +0000

Peter (2): There are several notions of constructive finiteness. The one I wrote about is Kuratowski’s notion, and it may be characterized as the free join-semilattice completion. The one you suggest ammounts to subsets of finite sets, and that is called subfinite (what is the categorical characterization?). You could restrict subfinite subsets to get a smaller (and more reasonable) class, for example the subsets of finite sets defined by `Pi_1^0` predicates are a possibility. Or you could take the intersection of the Kuratowski finite subsets with the compact subsets (with respect to a chosen dominance). There are indeed many possibilites, but the Kuratowski finite sets are the generally accepted “default” notion.

By: Peter LeFanu Lumsdaine

Peter LeFanu Lumsdaine — Fri, 11 Sep 2009 05:17:17 +0000

This definition has always slightly puzzled me: not because it’s too “un-classical”, but because it’s not “un-classical” enough — in particular, I always get nervous when a definition gives 0 special treatment. For instance, as you point out, we can prove “every set is empty or inhabited”, i.e. “every finite set has either at most 0 elements or at least 1″, but we can’t prove analogous things like “every finite set has either at most 1 element, or at least 2″ (we can see that this would imply excluded middle by considering the set {T,p} for any truth value p).

On a more practical note, I’d have expected that the same sort of phenomena (like looking at roots of polynomials) that give us finite sets in which we can’t decide whether or not there are multiple distinct elements would also give us “finite” sets in which we can’t decide whether there’s any inhabitant at all?

So I’d rather see a more relaxed definition, something like:

A set S is finite if there is some surjective _partial_ map from some natural number onto S.

Is there a good way to see why this would be unsatisfactory, or why my worries about the standard definition are unfounded?

By: Bob Harper

Bob Harper — Wed, 09 Sep 2009 22:24:56 +0000

It might be nice to compare various notions of “finite set” from a constructive viewpoint, ie with respect to their computational content. There’s a huge (computational) difference between saying “here’s a set and a natural number n such that i can map the set injectively into 0..n-1″, and “… i can map 0..n-1 onto the set”, and “no proper subset of this set can be mapped bijectively to this set”.