The real numbers and the Von Neumann Universe

Question

So I'm going to prefix this question by saying that I probably don't have a great understanding of what I'm asking.

We build the cumulative hierarchy as follows:

$V_0=\emptyset$

For every $\alpha$, $V_{\alpha+1}=\mathcal{P}(V_\alpha)$

If $Lim(\lambda)$, then $V_\lambda=\bigcup _{\alpha<\lambda} V_\alpha$

We then define the Von Neumann universe to be the class $V=\bigcup_{\alpha} V_{\alpha}$

Once we have done this we can prove various things about this such as:

Every set is in some $V_\alpha$

Intuitively we are supposed to picture this as the ordinal numbers being a vertical line starting at $\emptyset$ and going upwards. Then for each $\alpha$ the collection of sets of that rank are horizontal lines, so we get a sort of V shaped picture.

What I am unsure about (and maybe this is a ridiculous question) is where in this construction the real numbers are? We have that the natural numbers are $\omega$- the first transfinite ordinal and that $\omega_1$-the supremum of all countable ordinals but I am unsure where the real numbers come in?

Thanks for any help (sorry if the question is nonsense)

Answer 1

The real numbers show up in $V_{\omega+n}$ for some small finite $n$ whose precise value is sensitive to the exact details of how you choose to construct the reals.

Even before you have to choose between Dedekind cuts and Cauchy sequences, the rationals are usually constructed as (infinite) equivalence classes of pairs of integers, and the integers themselves as (infinte) equivalence classes of pairs of naturals. Each of these constructions can only happen after $V_\omega$ and contribute a level to $n$, and then the Kuratowski pairs you use in the next construction take a few additional levels to show up.

However, if you tune your constructions specially for the reals to exist early in the Von Neumann hiearchy, you can use canonical representatives rather than equivalence classes to represent integers and rationals. Then every rational is represented by a hereditarily finite set, and then $\mathbb Q$ itself as well as all its subsets will be present already in $V_{\omega+1}$ and you can have $\mathbb R\in V_{\omega+2}$ by Dedekind cuts.

Note, however, that for many set theorists "the reals" tend to mean simply $\mathcal P(\omega)$ rather than $\mathbb R$, and $\mathcal P(\omega)$ certainly arises already in $V_{\omega+2}$.

In any case, you cannot get $\mathbb R$ earlier than $V_{\omega+2}$, because every member of $V_{\omega+1}$ is at most countable.

Answer 2

The real numbers are not an intrinsic object to the universe of set theory. We have a good way of constructing them from the natural numbers, but actually every set of size continuum can be made into the real numbers.

In particular we have that $V_{\omega+1}$ is of size continuum, so in $V_{\omega+2}$ you already have a set of size continuum which can function as the real numbers (e.g. $\mathcal P(\omega)$ ordered by $A\prec B\iff\min(A\Delta B)\in A$).

If you wish to compute another construction of the real numbers, then you can do that manually. Suppose we wish to think of the real numbers as Dedekind-cuts, i.e. subsets of $\mathbb Q$, so we need to find when $\mathbb Q$ enters the universe; but again we have the same problem. What is $\mathbb Q$? Well, we can think of it as a quotient of $\mathbb Z\times\mathbb Z$, so again... when does $\mathbb Z$ enters the universe? Well, $\mathbb Z$ is a quotient of $\omega\times2$.

Let us consider the following rules:

Suppose that $A$ has rank $\alpha$. We know that pairs from $A$, $\langle a,b\rangle=\{\{a\},\{a,b\}\}$, which means that $A\times A\subseteq\mathcal{P P}(A)$, so $A\times A$ has rank of $\underline{\alpha+3}$.

Quotients of $A\times A$ are subsets of $A$, though, so they have rank of $\alpha+1$. Now you need to sit and calculate, if $\omega$ has rank $\alpha$ how do we get to $\mathbb R$?

Furthermore, if you also want the real numbers alongside with additions and other operations, you need to go higher as well, because those operations are only generated at higher stages.

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For further reading:

1. Formalising real numbers in set theory
2. In set theory, how are real numbers represented as sets?

Answer 3

The reals are certainly in $V_{\omega+2}$, no? $V_{\omega+1}$ is the first set to contain an infinite set, i.e $\omega$ itself, and since the cardinality of $\mathbb{R}$ is that of $\mathcal{P}(\omega)$, $\mathbb{R} \in V_{\omega+2}$.

Or are you asking about $V_{\gamma}$ where $\gamma$ is the first ordinal whose cardinality is the same as that of $\mathbb{R}$? In that case, there can be no precise answer - since the continuum hypothesis is undecidable in ZFC, you don't know how many cardinals lie between $\omega$ and $\gamma$.