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

1. A well-ordering of the set of natural numbers is a family of pairs of natural numbers, so there are at most \(2^{\aleph_0}\) of those. On the other hand, any permutation of natural numbers gives rise to a well-order, and there are \(2^{\aleph_0}\) of those. Hence there are continuum many well-orderings of the natural numbers.

2. Formally, yes: if we can prove that there are no cardinals satisfying the property \(\varphi(x)\), then we can prove "for every pair of cardinals \(x,y\), if \(\varphi(x)\), then \(x>y\)". This is an example of. If this isn't what you were asking about, feel free to clarify the question.