So, you may have heard about these things called countable elementary submodels. You may have heard that they work like magic and do all sorts of amazing things. “Mathematical voodoo” some might say. “Witchcraft!” others declare. Hearing this you become intrigued and set out to harness this black power. You quickly realize that there are very few places to learn this dark art; the protectors of this knowledge don’t want it leaking out.
Here I hope to lay out the essential things you need to know (and omit the things you don’t need to know) so that you can start using countable elementary submodels. I am going to lay out as little of the machinery as possible and display only the relevant applicable facts you will need for most proofs involving elementary submodels.
1. A Countable Submodel of What?
The universe of all sets is a ‘model’ for set theory, but it is too big. If we did have a model for set theory we would know that there is a countable submodel of it, by Lowenheim-Skolem. Of course we can’t assert that set theory has a model as this would be equivalent to asserting the consistency of set theory. The clever way around this is to realize that any proof in mathematics only ever uses finitely many axioms of set theory and references only finitely many specific sets. It is always possible to find a model of those finitely many axioms and special sets. (Aside, for those of you who have seen this before, why doesn’t this violate the compactness theorem? It’s tricky.) Here will be our copy of the universe, just for a given proof, and we will take a countable submodel of , not . This is where the language “Take a large enough fragment of ZFC” comes from.
As it turns out there is a class of sets that we usually draw from. We usually take to be a set , where is a cardinal and is the set of all sets hereditarily of cardinality less than . This doesn’t really matter at all. So don’t fret about this. Continue reading A Practical Guide to Using Countable Elementary Submodels