Big, little, nothing, everything

by Alexander Kharazishvili IMAGE/Duck Duck Go

No finite set is bigger than everything, and so every finite set is smaller than something. If one object is added to a small finite set, it gets bigger, but stays small.

Wait a minute. Could that be right?

Yes, absolutely. A world in which every finite set is small is perfectly consistent. This paradox is not so trivial as it might seem, if only because it is no paradox at all. In a proper paradox, two propositions must be in conflict. The set of all sets that are not members of themselves both is, and is not, a member of itself. This is Russell’s paradox. The conflict is plain, so such a set cannot exist.

Small sets are an irritant to intuition. In general set theory and model theory, ideals and filters are used to distinguish the small and large subsets of a ground set. The elements of an ideal are small. With filters, it is the other way around. An ultrafilter is a filter made maximal by inclusion. It includes everything that is large; all other subsets are small. The axiom of choice establishes the existence of nontrivial ultrafilters in every infinite ground set. These distinctions collapse in the case of finite sets, because any ultrafilter in a finite set is determined by a singleton.

In 1923, John von Neumann proposed to define the natural numbers by beginning with nothing. He began with the empty set Ø, and then argued that since nothing is nothing, 0 = Ø. His inductive definition of the natural numbers followed: n + 1 = n ? {n}. This implies that any natural number n coincides with the family of strictly smaller natural numbers. 1 is the one-element set {Ø}, 2 is the two-element set {Ø, {Ø}}, and so on up.

Finite sets may now be defined by means of the natural numbers. A set E is finite if there exists a one-to-one correspondence between E and some natural number n. Finiteness is absolute and remains invariant in all reasonable mathematical models. If the natural numbers may be used to define the concept of a finite set, the argument goes in reverse. It was Wac?aw Sierpi?ski who showed that it was possible to define the finite sets without appealing to natural numbers. It is as simple as 1, 2, 3. A class K of sets is admissible if

the empty set Ø belongs to K,
every one-element set belongs to K; and
if X ? K and Y ? K, then X ? Y ? K.

A set Z is called finite if Z belongs to all admissible classes of sets. The natural numbers express the cardinalities of Sierpi?ski’s finite sets. This definition relies on the concept of a proper class of sets. This is not a concept defined in Zermelo–Fraenkel set theory. Come to think of it, how can one element sets be a part of a definition abjuring the natural numbers?

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