Which ordinals can be "mistaken for" $\aleph_1$?

Question

I've just finished working my way through Weaver's proof of the consistency of the negation of the Continuum Hypothesis in his book Forcing for Mathematicians. One of the key points in this proof is that if $\mathrm{M}$ is a countable transitive model of ZFC, then the ordinal numbers in $\mathrm{M}$ are absolute, i.e. they are necessarily identical to the ordinal numbers in our larger universe, whereas the cardinal numbers in $\mathrm{M}$ are relative, meaning that, for instance, $\aleph_1^M$, or the first ordinal which $\mathrm{M}$ "thinks is uncountable", may be (and in this case actually is) only countable in the outside model.

Since $\mathrm{M}$ only contains countable ordinals, it must be "mistaking" some countable ordinal for $\aleph_1$. What do we know about which countable ordinals can serve as $\aleph_1$ for a countable transitive model of ZFC? Clearly $\omega$ cannot, since $\aleph_0$ is absolute. But which ordinals can be mistaken as $\aleph_1$? In particular:

• Can $\omega^2$ be mistaken for $\aleph_1$? Can $\omega^\omega$?
• What is the smallest ordinal that can be mistaken for $\aleph_1$?
• What properties must $\aleph_1^M$ have as an ordinal? (For instance, my intuition tells me that it must be a limit ordinal, not a successor ordinal.)

Can someone either sketch an answer to one or more of these questions (so that I can get an idea of how one goes about answering them), or point me to another resource that does? Thanks!

Answer 1

This is a very natural question, but one which - in my opinion - isn't going to have a satisfying answer: there's good heuristic evidence that no snappier description of this property than its definition exists. I'll use the phrase "$\omega_1$-like" instead of "can be mistaken for $\aleph_1$."

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First, it's easy to check that $\omega^2$, $\omega^\omega$, $\epsilon_0$, and even "big" countable ordinals like $\omega_1^{CK}$ (= the first "non-computable" ordinal) have absolute definitions. So they aren't $\omega_1$-like. Similarly, being a successor ordinal is absolute, and since $\mathsf{ZFC}$ proves that $\omega_1$ isn't a successor ordinal no successor ordinal can be $\omega_1$-like.

In fact, letting $\eta$ be the least $\omega_1$-like ordinal (and working in $\mathsf{ZFC}$ + "$\mathsf{ZFC}$ has a transitive model" to guarantee its existence) there is a general obstacle here to any simple characterization of this $\eta$:

$(*)\quad$ There is no first-order theory $T$ such that $\eta=\min\{\alpha:L_\alpha\models T\}$.

Here $L$ is Godel's constructible hierarchy. It's worth noting that in fact $(*)$ can be significantly strengthened; for example, we can replace "$L_\alpha\models T$" with "$L_{\alpha^2}\models T$," and so on, without changing the argument.

We can prove $(*)$ by applying downward Lowenheim-Skolem (+ the Mostowski collapse) run inside $\mathsf{ZFC}$: in any model $M$ of $\mathsf{ZFC}$, there is a countable-in-$M$ transitive set (necessarily of countable height!) which $M$ thinks has the same first-order theory as $L_{\omega_1}$.

Basically, $\eta$ represents a very complicated phase transition. By contrast, something like $\omega_1^{CK}$ - while extraordinarily huge by most standards - is simple to detect in this way: $\omega_1^{CK}=\min\{\alpha:L_\alpha\models$$\mathsf{KP}$$\mathsf{+Inf}\}$.

Effectively, $(*)$ and its strengthenings rule out any simple description of $\eta$ other than its a priori definition; it's just so complicated that we can't hope for something much more concrete.

That said, to get a sense of what you hit along the way to $\eta$, see this summary paper by Madore (note that each ordinal in that paper has an absolute definition). The ordinal $\eta$ here is entry $2.23$ in Madore's list, the third-to-largest ordinal he considers.