What do you do for a living, and why are they paying you not to understand this?
Is the empty set countable? (Yes.)
Dictionary:
nat·u·ral num·bers
the positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero as well
Set theory:
> Equivalently, a set S is countable if there exists an injective function f : S → N from S to N; it simply means that every element in S corresponds to a different element in N.
Defining N is usually done via a successor set, on which case 0 makes no sense to include.
Standard construction of ordinals is that each ordinal is the set of all its predecessors. (0 has no predecessors , hence 0 is the empty set.) (And so finite ordinals have the same ordinaliity as cardinality).
Birthdays are clearly 1 indexed, the first birthday is indexed with 1, and not with 0.
Birthday[0] gives you an out of range exception, since there is no birthday called 0th. Birthdays are [first,second,..] indexed from 1: brithday[1] = first, birthday[2] = second, and so on. That's what indexing a series from 1 means.