oh, yes, i think this is what is known as the continuum hypothesis. it was shown to be independent of ZFC, but is usually taken to be false, if i recall correctly.

yes. so the mapping X |--> X+1 is not injective, which is a requirement for the succesor map in peano s axioms. is the injectivity of the successor map required to prove the induction principle? i wouldn t be surprised.

so then the successor map on the ordinals is injective?

i see. proof by weak induction applies only to the range of the successor map. anything that is not in that range (like 0 and \omega) have to be done seperately and explicitly.

so what exactly do you mean when you say "limit step"? i am guessing that it is that stronger statement of the strong induction principle? i.e. that your statement is true for all numbers less than n implies it s also true for n, which would be explicitly required if n is not the successor of anything. is this correct?

yeah, that was pretty good. thanks.