Bare indicative conditionals are bewildering, but they become
surprisingly well-behaved if we add an 'else' clause.
Intuitively, 'if A then B' doesn't make an outright claim about the
world. It says that B is the case if A is the case – but what
if A isn't the case?
An 'else' clause resolves this question. 'If A then B else C' makes
an outright claim. It says that either B or C is the case, depending on
whether A is the case. That is: the world is either an A-world, in which
case it is also a B-world, or it is a ¬A-world, in which case it is a
C-world. For short: (A∧B)∨(¬A∧C).
Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.