For those who are not familiar with the notation, A! is the set of all permutations of the set A (permutation being defined as a bijection A~A from the set A to itself). There is a theorem in my notes which says "For cardinal number n it makes says to define n! = |A!| for some set A such that |A| is n." I'm trying to prove this theorem (missed my lecture :P) and I don't understand what it means by "it makes sense"? Surely if we're defining n! (where n is a cardinal number, not a natural number) then it can be anything we want and therefore it will make sense because we have defined it to.