User blog comment:Simplicityaboveall/Simplicity/@comment-1605058-20141223184654/@comment-1605058-20141226083954

Part 1 really is a mistake. What you have provided is the very definition of a function.

The fact that $$\aleph_0+1=\aleph_0$$ is a standard fact from cardinal arithmetic. The definition of sum of cardinals $$\alpha+\beta$$ is, by definition, cardinality of union of disjoint sets, first of these having cardinality $$\alpha$$ and second one cardinality $$\beta$$. In this case, first set is $$\{1,2,3,...\}$$, of cardinality $$\aleph_0$$, and second one is $$\{0\}$$, of cardinality $$1$$. Thus, by definition of cardinal addition, cardinality of $$\{0,1,2,3,...\}$$ is $$\aleph_0+1$$. But in the world of cardinals addition behaves weirdly, and we have that $$\aleph_0=\aleph_0+1=\aleph_0+2=...=\aleph_0+\aleph_0$$ and so on. So, it's incorrect to say that set of nonnegative integers doesn't have cardinality $$\aleph_0+1$$. One could even say that this set has cardinality $$\aleph_0+\aleph_0+42$$ and not be mistaken, because mentioned sets do have mentioned cardinalities.