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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]
| 13038 | Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen] |
| 13030 | Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen] |
| 13029 | Set Existence: ∃x (x = x) [Kunen] |
| 13031 | Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen] |
| 13034 | Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen] |
| 13032 | Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen] |
| 13033 | Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen] |
| 13036 | Choice: ∀A ∃R (R well-orders A) [Kunen] |
| 13037 | Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen] |
| 13039 | Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen] |
| 13040 | Constructibility: V = L (all sets are constructible) [Kunen] |