Category:Axioms/Axiom of Replacement
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This category contains axioms related to Axiom of Replacement.
Set Theory
For every mapping $f$ and subset $S$ of the domain of $f$, there exists a set containing the image $f \sqbrk S$.
More formally, let us express this as follows:
Let $\map P {x, z}$ be a propositional function, which determines a mapping.
That is, we have:
- $\forall x: \exists ! y : \map P {x, y}$.
Then we state as an axiom:
- $\forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x, y} }$
Class Theory
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.
Pages in category "Axioms/Axiom of Replacement"
The following 11 pages are in this category, out of 11 total.
R
- Axiom:Axiom of Replacement
- Axiom:Axiom of Replacement (Class Theory)
- Axiom:Axiom of Replacement (Set Theory)
- Axiom:Axiom of Replacement/Also known as
- Axiom:Axiom of Replacement/Also presented as
- Axiom:Axiom of Replacement/Class Theory
- Axiom:Axiom of Replacement/Class Theory/Formulation 1
- Axiom:Axiom of Replacement/Class Theory/Formulation 2
- Axiom:Axiom of Replacement/Class Theory/Formulation 3
- Axiom:Axiom of Replacement/Set Theory