The Oldest Definition
Frege and Bertrand Russell each proposed the following definition. Informally, each natural number n is defined as the set whose members each have n elements. More formally, a natural number is the equivalence class of all sets under the equivalence relation of equinumerosity. This may appear circular but is not.
Even more formally, first define 0 as (this is the set whose only element is the empty set). Then given any set A, define:
- as
σ(A) is the set obtained by adding a new element y to every member x of A. This is a set-theoretic operationalization of the successor function. With the function σ in hand, we can say 1 =, 2 =, 3 =, and so forth. This definition has the desired effect: the 3 we have just defined actually is the set whose members all have three elements.
This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theory ZFC and related systems, because in such systems the equivalence classes under equinumerosity are "too large" to be sets. For that matter, there is no universal set V in ZFC, under pain of the Russell paradox.
Hatcher (1982) derives Peano's axioms from several foundational systems, including ZFC and category theory. Most curious is his meticulous derivation of these axioms from the system of Frege's Grundgesetze using modern notation and natural deduction. The Russell paradox proved this system inconsistent, of course, but George Boolos (1998) and Anderson and Zalta (2004) show how to repair it.
Read more about this topic: Set-theoretic Definition Of Natural Numbers
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