Axiom:Axiom of Choice
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Axiom
For every set, we can provide a mechanism for choosing one element of any non-empty subset of the set.
- $\forall x \in a: \exists P \left({x, y}\right) \implies \exists y: \forall x \in a: P \left({x, y \left({x}\right)}\right)$
That is, one can always create a choice function for selecting an element of any set.
Alternative Version
Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of non-empty sets, indexed by $I$ which is also non-empty.
Then there exists a family $\left \langle {x_i} \right \rangle_{i \in I}$ such that:
- $\forall i \in I: x_i \in X_i$
That is, the Cartesian product of a non-empty family of non-empty sets is non-empty.
Comment
Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to transfinite sets this axiom leads to non-intuitive results, notably the famous Banach-Tarski Paradox.
For this reason, the Axiom of Choice (often abbreviated AoC or AC) is treated separately from the rest of the Zermelo-Fraenkel Axioms.
Set theory based on the Zermelo-Fraenkel axioms is referred to ZF, while that based on the Z-F axioms including the AoC is referred to as ZFC.
Also see
- Results about the Axiom of Choice can be found here.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 15$: The Axiom of Choice
- Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents
- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
