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In ZFC if there is a bijection between every set and some ordinal, why even use sets in the first place? Why not just assume everything is an ordinal? Come to think of it how do we know certain sets outside of the ordinals even exist in ZFC?

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Your last question is easy. Power sets of ordinals can be well-ordered, but they are not ordinals themselves (well, except for the cases of $\mathcal P(\varnothing)$ and $\mathcal P(\{\varnothing\})$ anyway).

But, ordinals are not enough, because we are interested in relations on sets, some structure. Just because the real numbers can be well-ordered it doesn't mean that their natural structure is somehow compatible with the natural structure of the ordinals; we need more than just that.

So that would be sets of ordinals. And indeed, a classical theorem of Balcar and Vopenka tells us that two models of ZFC with the same sets of ordinals are equal.

All in all, this is used often. For example, in many cases when one talks about iterated forcing, or some sort of structures, one starts with the implicit assumption that the underlying set which is being structured is an ordinal, or a set of ordinals.

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  • $\begingroup$ Thanks, it says I have to wait 5 minutes before accepting. Also two quick last things. When one says a set $S$ has cardinality $\kappa$ and writes $|S|=\kappa$. What exactly is $|.|$? I've always assumed it was some kind of class function or something? If it is the case, it can't have fixed points right? Since otherwise there would be some ordinal $\alpha$ with $|\alpha|=\alpha$ which doesn't seem right. A cardinal can't be an ordinal right? $\endgroup$ – user3865123 May 7 '18 at 7:36
  • $\begingroup$ It denotes that: (1) $\kappa$ is an initial ordinal, and (2) there is a bijection between $S$ and $\kappa$. And yes, $|\kappa|=\kappa$, when $\kappa$ is an initial ordinal. $\endgroup$ – Asaf Karagila May 7 '18 at 7:40
  • $\begingroup$ Wait so the cardinals are the ordinals?? $\endgroup$ – user3865123 May 7 '18 at 7:40
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    $\begingroup$ @user3865123 No, not the ordinals. But all cardinals are ordinals (by definition). $\endgroup$ – Tobias Kildetoft May 7 '18 at 7:43
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    $\begingroup$ @user3865123: Jech is not suitable for introduction, it is an intermediate book. Try Enderton's "Elements of Set Theory". $\endgroup$ – Asaf Karagila May 7 '18 at 8:18

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