Minimal Infinite Successor Set Exists

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Theorem

There exists a minimal infinite successor set $\omega$ that is a subset of every other infinite successor set.


Proof

From the Axiom of Infinity, there is a set $S$ such that:

$\varnothing \in S$
$x \in S \implies x^+ \in S$

where $\varnothing$ denotes the empty set and $x^+$ is the successor set of $x$.

That is, there exists an infinite successor set.

Next, by the Axiom of Subsets, the minimal infinite successor set $\omega$:

$\omega := \displaystyle \bigcap \left\{{S' \subseteq S: \text{$S'$ is an infinite successor set}}\right\}$

exists.


It remains to be shown that if $T$ is an infinite successor set, then $\omega \subseteq T$.

By Intersection of Infinite Successor Sets, $S \cap T$ is an infinite successor set.

Moreover, by Intersection is Subset, $S \cap T$, so that:

$S \cap T \in \left\{{S' \subseteq S: \text{$S'$ is an infinite successor set}}\right\}$

from which we conclude by definition of $\omega$ and Intersection is Subset: General Result that:

$\omega \subseteq S \cap T \subseteq T$

as desired.

$\blacksquare$


Sources