What is $\omega_\omega$?

Question

We all know $\omega_0$ the first countably infinite ordinal. And of course $\omega_1$, the first uncountably infinite one. But what is $\omega_\omega$?

This seems like a beast. Is its existence provable? What is this ordinal like?

Answer 1

The existence of $\omega_\omega$ - and in general of $\omega_\alpha$ for any ordinal $\alpha$ - is indeed provable in ZFC. The key axiom (scheme) is replacement (in particular, choice plays no role). The proof roughly goes like this:

• Suppose otherwise, and let $\alpha$ be the least ordinal such that "$\omega_\alpha$ doesn't exist" (phrased appropriately).

• By Replacement, the class of ordinals in bijection with $X$ forms a set, for any set $X$ (this is a multi-step argument); so we can look at the set $S$ of ordinals in bijection with $\omega_\beta$ for some $\beta<\alpha$ (EDIT: and the finite ordinals).

• Now $S$ is an ordinal (any downwards-closed set of ordinals is an ordinal), and it's not hard to show that $S$ doesn't inject into any $\omega_\beta$ for $\beta<\alpha$ but every proper initial segment of it does inject into some $\omega_\beta$ with $\beta<\alpha$.

• So $S=\omega_\alpha$. Oops.

It turns out that replacement is necessary: e.g. $V_{\omega+\omega}$ is a model of ZC (= ZFC without replacement), but $V_{\omega+\omega}$ doesn't even think $\omega_1$ exists!

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As to how large $\omega_\omega$ is, that's a bit hard to answer. One key fact is that it is (as tomasz comments) consistent with ZFC that $\vert\mathbb{R}\vert>\omega_\omega$. Indeed, it turns out that pretty much the only thing one can prove about the continuum in ZFC is that it is uncountable and has uncountable cofinality. So, e.g., it could be $\aleph_1$ (this is the continuum hypothesis) but it can't be $\aleph_0$, and it can't be $\aleph_\omega$ (= $\omega_\omega$) since that has countable cofinality (and indeed one of the most important facts about $\omega_\omega$ is that it is the least infinite singular cardinal), but it could be e.g. $$\aleph_{\omega^{\omega^\omega}+\omega^2\cdot 37+12}.$$

One of the major themes of modern set theory is that the ZFC axioms - even plus large cardinals! - are absolutely terrible about proving things about cardinal exponentiation. On the other hand, one of Shelah's fundamental contributions to set theory is that ZFC can prove some surprising things about cardinal exponentiation: for example, that $$\mbox{If $2^{\aleph_n}<\aleph_\omega$ for all $n\in\omega$, then $2^{\aleph_\omega}<\aleph_{\omega_4}$.}$$ (The hypothesis there can be abbreviated as "$\aleph_\omega$ is a strong limit cardinal.") No, we don't know if "$4$" is optimal, there.