Single Idea 13028

[catalogued under 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII]

Full Idea

Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion.

Gist of Idea

Replacement was added when some advanced theorems seemed to need it

Source

report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8

Book Reference

-: 'Journal of Symbolic Logic' [-], p.489


A Reaction

Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'.

Related Ideas

Idea 15933 Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]

Idea 15945 Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]