Comparing between $\omega_2$ and $\omega_1$ My question, it seems very basic about cardinal numbers. I know $\omega_1$ is the first uncountable regular cardinal. Also, $\omega_2$ is the second uncountable regular cardinal.
I did not know how I can envision the difference between them. In other words, what are things that that might be achieved for $\omega_1$ but not for $\omega_2$ and so on.
 A: I suppose I'll give this a stab after all. For me personally, there are two differences between $\omega_1$ and $\omega_2$ which leap reflexively to mind whenever I consider them in the same thought.

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*At an elementary level, there is a difference in the structure of "natural" stationary sets on each cardinal. In line with Jason Zesheng Chen's comment above, $\mathsf{ZF+AD}$ proves that the club filter on $\omega_1$ is an ultrafilter. By contrast, consider $$A=\{\alpha\in\omega_2: cf(\alpha)=\omega\}\mbox{ and }B=\{\alpha\in\omega_2: cf(\alpha)=\omega_1\}.$$ As long as $\omega_1$ is regular (e.g. assuming countable choice) $A$ and $B$ are stationary but clearly $A\cap B=\emptyset$.


*On a more sophisticated level, the related structures $H_{\omega_1}$ and $H_{\omega_2}$ are quite different, model-theoretically speaking. Elements of $H_{\omega_1}$ are "morally equivalent" to real numbers, and so the first-order theory of $H_{\omega_1}$ is relatively immune to set-theoretic shenanigans, especially granting large cardinals. We have no such tameness in $H_{\omega_2}$, however; for example, the continuum hypothesis can be turned on or off as desired by forcing and is equivalent to a first-order sentence in $H_{\omega_2}$. Indeed, the search for a theory to "tame" $H_{\omega_2}$ analogous to projective determinacy's impact on $H_{\omega_1}$ is a very difficult one. While much more complicated than the previous bulletpoint, this is actually more reflexive to me since I use the relative tameness of $H_{\omega_1}$ (or its moral equivalents) quite frequently in my own work.
But these are more technical differences: why should one care about natural stationary sets or model-theoretic tameness? Ultimately we're left with the bare fact that these ordinals are rather technical ($\omega_2$ especially), and ultimately a good picture of them will only emerge after (or at best, during) the process of learning how to work with them.
