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I am trying to understand (self-study) a few things from set theory. One particular claim I encountered is the following: "Can we always rank the cardinalities of any two sets? The answer is affirmative, and when equipped with the Well-Ordering Principle, not difficult to prove."

They follow it up with an exercise question: Show that, for any two sets $A$ and $B$, we have either $A\succ_{card} B$ or $B\succ_{card} A$.

I would like to get clues on how to approach this question. I know the Well-Ordering Principle.

Juanito
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  • @Asaf: The question is not easily searchable, as in, I could find it, possibly due to the notations being used. Should I delete my question given it is a duplicate? – Juanito May 29 '17 at 06:51
  • No reason to delete your question. Also, you have an upvoted answer, so you can't delete it anyway. – Asaf Karagila May 29 '17 at 06:52

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You should prove that if $A$ and $B$ are well-ordered sets, then one is order-isomorphic to an initial segment of the other. The key is transfinite recursion.

Angina Seng
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