Franklin's Notes


Hartogs Number

Given a set $X$, there exists an ordinal $\alpha$ such that $|\alpha|\not\leq |X|$, or equivalently such that $\alpha$ cannot be injected into $X$. Since every set of ordinals is well-ordered by $\subset$, we therefore have that there exists a least ordinal that cannot be injected into $X$, which we will denote $\aleph(X)$. It is necessary that $\aleph(X)$ be a cardinal and therefore a limit ordinal, for otherwise some ordinal smaller than it could be injected into $X$.

set-theory

ordinals

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