Relation Contains Mapping is Equivalent to AoC

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation on $S \times T$.


Then:

there exists a mapping $f \subseteq \RR$ whose domain is the same as the preimage of $\RR$

if and only if

the axiom of choice holds.


Proof




Axiom of Choice

This theorem depends on the Axiom of Choice.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.


Sources