How do you generate the numbers from an empty set? In a short 2007 Scientific American article, the former Harvard mathematician and author Dr Robert M Kaplan stated that:

in mathematics we can generate all numbers from the empty set

I've not been able to find the proof of that statement. I assume he means the natural numbers. I can see how you can get to the empty set by removing each number from a set of numbers, therefore, reverse that process. (Though I wonder if that applies to an infinite set?) But what is the start point? Is the start related to counting the span of the empty set? I.e. by counting the span of the empty set to get the first number then a proof can be shown that involves cardinality.
What is the straightforward proof that demonstrates that all numbers can be generated from the empty set?
 A: You can do better than just all natural numbers: you can construct every ordinal!
Nickname $\varnothing$ as $0$. For any $a$, let $a+1=a\cup\{a\}$. For any limit ordinal $\delta$, let $\delta=\bigcup\limits_{\alpha<\delta}\alpha$.
Hence, we've defined the ordinals as sets, just by starting with the empty set and building upwards.
Now how do we know that this is comprehensive? All we want to show is that every well-ordered set is in bijection with one of these ordinals. But that's a classic theorem you can find here.
(I'd be remiss to not mention that this construction is a bit circular, since we are assuming the existence of ordinals while constructing them, but if you want a proper treatment of this subject, there are plenty of texts for you.)
A: There is a standard set-theoretic method to use the empty set and its succession of iterated power sets to make a model of the natural numbers.
Once you have the naturals, you can model integers as pairs of natural numbers: $(a,b)$ represents the integer $a-b$.
Once you have the integers, you can model the rationals as pairs of integers: $(c,d)$ represents the rational number $c/d$.
Once you have the rationals, you can model the reals using Dedekind cuts.
Once you have the reals, you can model the complexes using a field extension by a root of $x^2 +1 = 0$ (so again, using pairs, $(e,f)$ representing $e + \mathrm{i}f$) or by constructing the algebraic closure of the reals.  (You get the same complex numbers either way, although the proof of equivalence is not a one-liner.)
Other sets of numbers, quaternions, octonions, et c., proceed similarly.  Cardinals and ordinals are addressed in the other answer(s) to this question.
