User blog comment:Alemagno12/Some set theory questions/@comment-1605058-20171205215012/@comment-1605058-20171206094242

I've written a longer reply, but Wikia has decided to not let me send it, so here is a TL;DR version because I am not going to write it all again.

@Alemagno: It seems that your statement is implying that it's the cardinals not satisfying \(\varphi(x)\) are the large ones. Whether you meant it or not, the statement you ask about is true, because both parts of it are vacuously true.

@Edwin: The reason we need vacuous truth is that it does come up when we want some "nontrivial" universal quantifications to be true. Consider for example the statement in the positive integers "for every \(n\), for every proper divisor \(m\) of \(n\), \(m<n\)". This is a true statement, but if we take \(n=1\), we get a statement "for every proper divisor \(m\) of \(1\), \(m<1\)". This is quantification over the set of proper divisors of \(1\), which is empty. We need this empty set quantification to be true in order for the original statement to hold.

This is also very closely related to how for implication \(P\implies Q\), if \(P\) is false, then the implication always holds.