2023-12-31 21:51:20 +01:00
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# Additional axioms
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## Axiom of choice
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> *Principle*: let $C$ be a collection of nonempty sets. Then there exists a map
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>
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>$$
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> f: C \to \bigcap_{A \in C} A
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>$$
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>
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> with $f(A) \in A$.
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>
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> * The image of $f$ is a subset of $\bigcap_{A \in C} A$.
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> * The function $f$ is called a **choice function**.
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The following statements are equivalent to the axiom of choice.
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* For any two sets $A$ and $B$ there does exist a surjective map from $A$ to $B$ or from $B$ to $A$.
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* The cardinality of an infinite set $A$ is equal to the cardinality of $A \times A$.
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* Every vector space has a basis.
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* For every surjective map $f: A \to B$ there is a map $g: B \to A$ with $f(g(b)) = b$ for all $b \in B$.
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## Axiom of regularity
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> *Principle*: let $X$ be a nonempty set of sets. Then $X$ contains an element $Y$ with $X \cap Y = \varnothing$.
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As a result of this axiom any set $S$ cannot contain itself.
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