Axiom:Axiom of Replacement/Class Theory/Formulation 1
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Axiom
For every mapping $f$ and set $x$ in the domain of $f$, the image $f \sqbrk x$ is a set.
Symbolically:
- $\forall Y: \map {\text{Fnc}} Y \implies \forall x: \exists y: \forall u: u \in y \iff \exists v: \tuple {v, u} \in Y \land v \in x$
where:
- $\map {\text{Fnc}} X := \forall x, y, z: \tuple {x, y} \in X \land \tuple {x, z} \in X \implies y = z$
and the notation $\tuple {\cdot, \cdot}$ is understood to represent Kuratowski's formalization of ordered pairs.
Also known as
The axiom of replacement is also known as the axiom of substitution.
Also see
- Results about the Axiom of Replacement can be found here.
Historical Note
The axiom of replacement was added to the axioms of Zermelo set theory by Abraham Halevi Fraenkel, and also independently by Thoralf Albert Skolem.
The resulting system of axiomatic set theory is now referred to as Zermelo-Fraenkel Set Theory.
Sources
- 1940: Kurt Gödel: The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory
- 2010: Elliott Mendelson: Introduction to Mathematical Logic (5th ed.): $4$ Axiomatic Set Theory
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 3$ The axiom of substitution: $A_8$