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!