Minimally Inductive Set Exists
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Theorem
There exists a minimally inductive set $\omega$ that is a subset of every other inductive set.
Proof
From the Axiom of Infinity, there is a set $S$ such that:
- $\O \in S$
- $x \in S \implies x^+ \in S$
where $\O$ denotes the empty set and $x^+$ is the successor set of $x$.
That is, there exists an inductive set.
Next, by the Axiom of Specification, the minimally inductive set $\omega$:
- $\omega := \ds \bigcap \set {S' \subseteq S: \text{$S'$ is an inductive set} }$
exists.
It remains to be shown that if $T$ is an inductive set, then $\omega \subseteq T$.
By Intersection of Inductive Sets, $S \cap T$ is an inductive set.
Moreover, by Intersection is Subset, $S \cap T$, so that:
- $S \cap T \in \set {S' \subseteq S: \text{$S'$ is an inductive set} }$
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
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers