I had proved that infinite cardinal is limit ordinals.
but want to prove the inverse is false.
any ordinal which is limit ordinal but finite cardinal?
2026-04-03 18:22:46.1775240566
Limit Ordinal but Finite Cardinal
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The converse of your statement is "A limit ordinal is an infinite cardinal". You can disprove this with a limit ordinal that is not a cardinal, like $\omega + \omega$. All the limit ordinals have infinite cardinality, but they are not all cardinals.