Strict Lower Closure of Limit Element is Infinite

Theorem

Let $A$ be a class.

Let $\preccurlyeq$ be a well-ordering on $A$.

Let $x \in A$ be a limit element of $A$ under $\preccurlyeq$.

Let $x^\prec$ denotes the strict lower closure of $x$ in $A$ under $\preccurlyeq$.


Then $x^\prec$ is an infinite set.


Proof

Let $x \in A$ be a limit element of $A$ under $\preccurlyeq$.

From Characterisation of Limit Element under Well-Ordering it follows that $x^\prec$ has no greatest element with respect to $\preccurlyeq$.

The result follows.

$\blacksquare$


Sources

  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering: Proposition $1.7$
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