Single Idea 13020

[catalogued under 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation]

Full Idea

The most characteristic Zermelo axiom is Separation, guided by a new rule of thumb: 'one step back from disaster' - principles of set generation should be as strong as possible short of contradiction.

Gist of Idea

The Axiom of Separation requires set generation up to one step back from contradiction

Source

report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.4

Book Reference

-: 'Journal of Symbolic Logic' [-], p.485


A Reaction

Why is there an underlying assumption that we must have as many sets as possible? We are then tempted to abolish axioms like Foundation, so that we can have even more sets!

Related Idea

Idea 18105 Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]