Existence of Set with Singleton Intersections with Disjoint Collection

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Theorem

Let $\mathcal C$ be a set of sets all of which are pairwise disjoint.


Then:

there exists a set $A$ such that $\forall S \in \mathcal C: A \cap S$ is a singleton

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