“By definition nothing does not exist, but the concepts we have of it certainly exist as concepts. In mathematics, science, philosophy, and everyday life it turns out to be enormously useful to have words and symbols for such concepts.”
“The null set is symbolized by ∅. It must not be confused with 0 (zero). Zero is (usually) a number that denotes the number of members of ∅. The null set denotes nothing, but 0 denotes the number of members of such sets, for example the set of apples in an empty basket. The set of these nonexisting apples is ∅, but the number of apples is 0.”
“A way of constructing the counting numbers, discovered by the great German logician Gottlob Frege and rediscovered by Bertrand Russell, is to start with the null set and apply a few simple rules and axioms. Zero is defined as the cardinal number of elements in all sets that are equivalent to (can be put in one-to-one correspondence with) the members of the null set. After creating 0, 1 is defined as the number of elements in all sets equivalent to the set whose only member is 0. [Should this be instead “∅”..?]. Two is the number of members in all sets equivalent to the set containing 0 and 1. Three is the number of members in all sets equivalent to the set containing 0, 1, 2, and so on. In general, an integer is the number of members in all sets equivalent to the set containing all previous numbers.”
Gardner, Martin. The night is large: collected essays 1938-1995. St. Martins Griffin (1996), pp 397, 398
The set, { }, has zero elements.
The set, { { } }, has one element.
The set, { { }, { { } } }, has two elements.
The set, { { }, { { } }, { { }, { { } } } }, has three elements.
The set, { { }, { { } }, { { }, { { } } }, { { }, { { } }, { { }, { { } } } } }, has four elements. …
By continuing this pattern one can build the entire set of natural numbers. To me, this is truly fascinating.