What techniques are available for "surgical adjustment" of models of set theory?

Question

Suppose I have a model $M$ of set theory (ZFC, or whatever). Let's say that I want to take a set $a$ out of it, and still have a model of set theory. For the sake of argument, say $a$ is one of the indefinable real numbers. Intuitively, $a$ is not "required" for $M$ to be a model of set theory; there are models of set theory without indefinable real numbers. So we ought to be able to take it out.

The naive approach is to take $M - a$ with the induced membership relation, and try to verify that it's still a model of set theory. But it isn't; e.g., $M$ must contain a singleton $\{a\}$, and in $M - a$, the singleton $\{a\}$ is now empty; so we have at least two empty sets, violating extensionality. Probably this is hardly even the beginning of our problems.

We run into similar problems if we try to add a single new set to the model (e.g., a new subset of the reals). To add a single new set $b$, we need to add a singleton $\{b\}$; we need to add all finite sets containing $b$; we need to modify sets produced by the separation axiom to include $b$ where appropriate; etc.

So in general doing this sort of fine-grained surgery on a model of set theory is a difficult problem, even when intuitively it should be possible. I am wondering what techniques are available for this. I understand that forcing is, broadly speaking, a method for adding more sets. The omitting types theorem is one way to "remove" sets.

Unfortunately, I believe that none of the methods I know apply to the problem I am working on. In particular, I don't think forcing applies because my problem is one of removing sets, not adding them; and I don't think omitting types applies because meeting its hypotheses essentially seems to amount to already having solved the problem I am trying to solve. So I am casting about for other methods.

Question: What other techniques are available for fine-grained addition and removal of sets to/from a model of set theory?

Thank you!

Answer 1

There is no "fine grained" tool that I know of, but there are several ways of removing sets nonetheless. There are some minor points to this, which I will discuss at the end of the post.

Let $M$ be a model of $\mathsf{ZF}$, we say that $N\subseteq M$ is an inner model of $M$ if $N$ is transitive (with respect to $M$), contains all the ordinals (of $M$) and satisfies the axioms of $\mathsf{ZF}$. For example the canonical inner model is $L$, Godel's constructible universe, which is the smallest inner model. It contained in any model of $\mathsf{ZF}$ and if two models have the same ordinals they have the same $L$.

There are other inner models, $HOD$, for example, the class of those which are hereditarily ordinal definable, is an inner model (which may or may not be equal to $L$). In some cases there are inner models defined from large cardinals by elementary embeddings; and in other cases we may have obtained the universe by a generic extension which means that is some inner model which was the ground model in the process of forcing.

Once we are inside a fixed universe of set theory, dealing with inner models can become slightly simpler. If $N$ is an inner model and $x\in M$ then $N(x)$ is the intersection of all inner models which contain both $N$ and $x$, this class is not empty because $M$ is an inner model of itself which contain both. One can also talk about $N[x]$ which is the inner model of all the things constructible from $N$ and $x$, and often the two notions coincide, but let's leave that for another time.

So how does that help us? Well, if $a$ is a "complicated enough", i.e. $a\in M\setminus L$, then $L$ is an inner model in which $a$ is not present. We could try and extend it by adding other sets, in case where it is possible.

For example if our universe was a generic extension of $L$ by adding two reals, then we can add just the one and obtain an intermediate model which is larger than $L$ but smaller than the full universe.

Similarly we don't really need to limit ourselves to $L$. If we can prove that $a$ is not present in some inner model we can do the same trick. Start with some $N$ and slowly add sets, if we want, and stop before we add $a$ again.

Two caveats:

1. We don't really have a surgical tool for removing "just this one element", because removing one set implies that we have to remove all the sets which include him, and all the sets which are not definable without it. The fact that there is a smallest inner model tells you that sometimes you can't remove sets. If your universe satisfies $V=L$ then there is no way to surgically remove sets.

2. Definability is a fussy concept here, when you say "undefinable real" most people think about some transcendental number which can't really be defined, but those exist even in models like $L$. So even if something is not definable in the "usual" sense of the word, it might still be necessary.

I will finish with with pointing out that there has been some work recently in what is known as set theoretical geology which explores inner models that the universe is a generic extension of. It may prove useful, although I'm not a 100% sure how.