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Single Idea 13034

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

Full Idea

Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y.

Gist of Idea

Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y)

Source

Kenneth Kunen (Set Theory [1980], §1.6)

Book Ref

Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.12

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]

The 5 ideas with the same theme [axiom saying the bijection of any set is also a set]:

13028 Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy]
13202 Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
18192 Do the Replacement Axioms exceed the iterative conception of sets? [Boolos, by Maddy]
13034 Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen]
15899 Replacement was immediately accepted, despite having very few implications [Lavine]