I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are limit ordinals. My question is: are there any ordinals that are not cardinals? Although I know that an ordinal describe ordering and a cardinal the size of a set, I am a bit confused here.


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    $\begingroup$ There might be better duplicates, but that was the first I found. $\endgroup$ – Asaf Karagila Feb 14 '15 at 23:57
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    $\begingroup$ The main point is that an infinite set can be well-ordered in many non-isomorphic ways. And while cardinals measure the size of a set when we ignore any structure on the set; ordinals measure how "long" is a particular structure on a set. $\endgroup$ – Asaf Karagila Feb 14 '15 at 23:58
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    $\begingroup$ Here are some threads where many words have been minced over cardinals and ordinals: one, two, and three. There are probably many more around here, but those are three that I found quickly enough. $\endgroup$ – Asaf Karagila Feb 15 '15 at 0:01