Modelling Infinite Structures with Atoms

(Link).

Author: Mikołaj Bojańczyk (University of Warsaw)

Conference: Workshop on Logic, Language, Information and Computation (WoLLIC 2013), 20-23 August, 2013, Darmstadt, Germany.

Journal reference: Lecture Notes in Computer Science, to appear.

Comments: The topic of this week's paper, from an author we've met twice before (here and here) is compactness. Compactness is the property that states that an infinite set of logical sentences is satisfiable (not contradictory) if and only if all the finite subsets of that infinite set are satisfiable. Some logics have the property, but others may not. The importance of this result may not be immediately obvious; it was first discovered by Kurt Gödel for first-order logic as a mere corollary to another theorem (completeness). However compactness went on to be a cornerstone principle of the mathematical field of model theory, which in a broad sense looks at how logical claims can be 'satisfied' by mathematical constructions.

Bojańczyk is interested in shifting first-order logic from its usual set-theoretic foundations to the world of nominal sets, the mathematical model for names in syntax that I've discussed in many places on this blog, e.g. here (in fact, as discussed in that link, Bojańczyk has a somewhat more general notion in mind than is usually meant by 'nominal sets', but such subtleties are orthogonal enough to the gist of this paper that we will pass by them). Making first-order logic nominal means that predicates can involve names, and disjunctions and conjunctions can be infinite, although only 'orbit-finite'.

Compactness is fundamentally a theorem about infinite and finite collections; in nominal sets such issues become a bit subtle. First, each element of a nominal set has what is called the finite support property, which means that it 'depends' on only finitely many names. This is a very helpful assumption in practice, and is motivated by the idea that real syntax can only ever involve finitely many names (even when the syntax might get infinitely large, as I discussed here). Moving up a level, we see that no non-trivial nominal set is finite. However there is a notion something like finiteness that does make sense for nominal sets - that of orbit-finiteness, which essentially says that the nominal set is finitely 'generated', rather like an infinite number of reflections in a hall of mirrors could be generated by a finite number of people standing in its middle.

With familiar notions of finiteness and infinity upended by the nominal sets model, getting a halfway sensible notion of compactness becomes quite difficult. It clearly makes no sense to state the theorem in terms of finite sets of sentences, because there are no interesting finite nominal sets. More surprisingly, the compactness postulate got by replacing the word 'finite' with 'orbit-finite' turns out to be false. In fact to validate that version of the theorem we initially need to jump outside of the nominal sets model entirely by axing the finite support property. This is by no means an illegitimate move but it does create worries, because sets with names but no finite support property are less relevant to computer science applications. However Bojańczyk finishes the paper by putting the rabbit back into the hat by defining what are called 'stratified models', more complicated constructions which have the finite support property but also satisfy compactness by 'approximating' the non-finitely supported models.

Update: Due to workload my next post will be a fortnight after this one.