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.
Have you ever tried to explain/inflict forcing on your non-set theory friends? Let me tell you it is hard. In my ongoing effort to try to explain everything to everyone here is my attempt at explaining the idea of forcing, without explaining forcing.
First, I direct you to this post which explains the simulation argument.
tl; dr – Imagine that our civilization could simulate (artificially) intelligent civilizations. They wouldn’t know that they are in a simulation and could also run their own simulations.
Reading this post reminded me about forcing. Here is my reply to the OP (who happens to be my brother-in-law), and I would appreciate feedback on this so that I can refine my analogy:
Cool. This illustrates a key observation that underlies a lot of the mathematics (set theory) that I do. It goes like this for the interested parties:
We run a simulation as you’ve described, but we make sure that the simulation only has “a small number of things in it”. Perhaps we have some sort of minimal simulation, like we don’t include the letter Z in their languages or something (Call this SIM1). Now we check that the simulation can come up with its own simulations. It can? Great! So now we know that “having the letter Z in your language” is not a requirement for coming up with simulations. Or we could add a whole bunch of new crazy letters (in say SIM2) and see if they can still run simulations. Lets say they can still come up with simulations. Continue reading Every Day I’m Simulating