Saturday, May 13, 2017

set theory - For any two sets $A,B$ , $|A|leq|B|$ or $|B|leq|A|$


Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally:



$$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$



If this statement is true, what is the proof ?


Answer




This claim, the principle of cardinal comparability (PCC), is equivalent to the Axiom of Choice.


If the Axiom of Choice is true then Zorn's Lemma is true and a proof of the PCC is a classical application of Zorn's Lemma.


If PCC holds then using Hartogs' Lemma it is quite easy to show that The Well-Ordering-Princple holds, which in turn implies (easily) the Axiom of Choice.


A complete presentation of these (and some other) equivalences, is treated in a rather elementary fashion but including all the details, in the book: http://www.staff.science.uu.nl/~ooste110/syllabi/setsproofs09.pdf starting on page 31.


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