I know that Freges and Russels approach to the foundation of math is not quite it.
But isn’t it enough to know what counting in essence is to have the foundation?
I mean, if we have recursion of functions via Y-combinator, we can form the idea of Peano numbers, for the def. of these is recursive, like expressed e.g. in haskell:
data Peano = Zero | Succ Peano
And this is the general concept of counting, which is need for everything math purports to be.
Or am I wrong? :]