User:Dfeuer/Definition:Cardinality/Scott

Definition
Assume that the Axiom of Foundation holds.

Let $S$ be a set.

Let $A$ be the class of all sets equivalent to $S$.

Then the cardinality of $S$ is the set of elements of $A$ of least rank.

Remarks
This definition gives a very different sort of cardinal number than the more common approach of defining the cardinality of a set as the least ordinal equivalent to it. The more common approach relies on the Axiom of Choice, while Scott's approach relies instead on the Axiom of Foundation.