What does the power set mean in the construction of Von Neumann universe?

Question

I'm recently learning mathematical logic and axiomatic set theory. I'm using GTM53 as the textbook for mathematical logic. In page 65 of GTM53, when explaining Skolem's paradox, there's a specific notification of power set that the meaning of the power set of a set should be determined by the model we choose, because the axiom of powers only states that: $$\forall x \exists y \forall z\left[\forall u(u \in z \to u \in x) \leftrightarrow z \in y\right],$$ and since the universal quantifier $\forall z$ ranges over the model, this sentence actually means that every set in the model which is the subset of $x$ belongs to $y$ (the power set of $x$). This indicates that if the model is "small", then the power set would also be small, and hence there exists a countable model that the power set of a countable set is also countable (in the viewpoint of meta-theory), which explains Skolem's paradox.

People say Von Neumann universe is a model of ZFC. However, in the counstruction of Von Neumann universe, we actually used power set symbol ($V_{\beta+1} = \mathcal P(V_\beta)$). If we must first determine a model to determine what the power symbol means? How can we say Von Neumann universe is model of ZFC as it uses a symbol that has not been defined?

Answer 1

First of all, what you are calling the von Neumann universe is nowadays more usually called the cumulative hierarchy. As von Neumann is also associated with another axiomatization of set theory (NBG), there is potential for confusion.

There is a formal definition of the power set in ZF, namely $y=P(x)$ means $$\forall z(z\in y\leftrightarrow z\subseteq x)$$ or expanding out the defined relation $\subseteq$, $$\forall z(z\in y\leftrightarrow \forall u (u\in z\to u\in x))$$ which as you can see is part of the axiom of power set you quoted. Using this defined relation, we can write the axiom: $$\forall x\exists y(y=P(x))$$ Suppose $M$ is a model of ZF. If $x$ and $y$ are elements of $M$, then $M\models y=P(x)$ just means that $x$ and $y$ satisfy that formal definition in $M$.

I'm belaboring this point to answer this part of your question: "How can we say Von Neumann universe is model of ZFC as it uses a symbol that has not been defined?" $P(x)$ has been defined formally, so within the model $M$ we can construct the cumulative hierarchy for that model.

Now, models of ZF have an interesting feature: you can also look at them from "outside". This assumes the existence of some outer universe containing the model. Not to get too deeply into the philosophical weeds, let's just call this the "real" universe of sets, with the scare quotes.

Corresponding to $x\in M$ we have an "actual" set in the "real universe". It's the collection of all $y$'s of $M$ for which $M\models y\in x$. We can form the "actual" power set of this "actual" set. The "actual" power set may be bigger than the power set in $M$, as you've mentioned.

To pursue this inner vs. outer discussion further we'd need better notation; also we'd need to bring up other issues. But I hope this clarifies things.

Answer 2

You seem to be badly mixing up the study of models of ZFC and the use of ZFC as a foundation for mathematics. These are two totally separate things. When you use ZFC as a foundation for mathematics, you assume the axioms of ZFC, and make definitions and prove theorems from those axioms. Among the definitions you can make are the power set operation, and the von Neumann hierarchy, and so on. You can also define things like the real numbers, and manifolds, and other mathematical structures of interest. All of this is perfectly well-defined and does not involve any discussion of models at all.

One type of mathematical structure you can then define is a model of ZFC. This is a set together with a binary relation such that it satisfies all the axioms of ZFC. Within such a structure, you can then "internalize" definitions like the power set operation. All that's going on in Skolem's paradox is that the power set operation you define in a specific model might not be the same as the true power set operation defined outside the model. Different models can have different power sets of the same set, and in this sense the power set "depends on a choice of model". But all of this is only talking about the internal power set operation of a model, not the external power set which you define when doing ordinary mathematics and aren't looking at a model. The external power set has nothing to do with models and you can use it to do math as you normally would (and define things like the von Neumann hierarchy).