Description of the Universe $V$

Question

For me, the concept "set" seams very ambiguous. This does not satisfy me because sets are used very often in mathematics, and so many questions in mathematics are not definite for me. I want to read an intuitive description of our universe of sets.

This universe $V$ is often explained in the context of the so called "von Neumann universe". Wikipedia says that the von Neumann universe is

often used to provide an interpretation or motivation of the axioms of $\mathsf{ZFC}$.

The von Neumann universe is the universe of sets one has at the back of one's mind when speaking about sets.

Can you explain the von Neumann universe so that I will have an intuition for this universe?

Here is the thing that disturbs me when I read some explanations of the von Neumann universe: They used ordinals. But ordinals are sets, and using ordinals to explain what sets are is circular somehow.

I am not speaking of a definition of the concept "set" and I am also not speaking about an axiomatization of $V$. I just want get an intuition about $V$.

How would platonists describe the universe?

Or, more specifically, what would Andreas Blass tell me if I ask him about $V$?

Answer 1

If you want to consider the "simple" foundational approach to theories like $\sf ZFC$, then sets are primitive objects and $V$ is a given universe to begin with.

The axioms of $\sf ZFC$ tell you what sort of properties $V$ and its $\in$ relation satisfy. For example, they tell you there is a set which is inductive, and they tell you that $\in$ is well-founded (as far as $V$ is concerned), and that if $X$ is a member of $V$, the there is a set in $V$ which is the power set of $X$ and so on.

What the von Neumann hierarchy gives you is the understanding that if $V$ is already given, then we can write this wonderful filtration of $V$ into a very nice hierarchy. Additional theorems like the reflection theorem also tell you more about this hierarchy and its deep connection with the structure of $V$ as a universe of sets.

But in either case you start with $V$ as a given concept, and a set is just something which belongs to $V$.

So what intuition can you get on $V$? Frankly, not a whole lot. It's a very complicated object, not to mention that different universes of sets can be very different, so you don't have nearly enough information about $V$ as it is. Does it satisfy this axiom or that axiom? Does it have large cardinals? What is the truth value of the continuum hypothesis in $V$? Are there Suslin trees? Is $V$ a set-generic extension of a smaller universe? All these are questions that just $\sf ZFC$ simply cannot answer. So as far as a Platonist go, you might get very different answers from one Platonist to another.

What respite can I offer you instead? I can suggest that you take comfort that in mathematics the intuition you have initially (or hoping to develop "immediately") is almost always wrong. It is through the understanding that this intuition fails us that we learn to work closely with the definitions (and axioms), and slowly we develop some sort of often-ineffable intuition about whatever it is that we work with. This can take several years to accomplish. But it is an extremely rewarding process to slowly realize that you understand what the hell is going on.