Minimal Infinite Successor Set forms Peano Structure
Theorem
Let $\omega$ be the minimal infinite successor set.
Let $\cdot^+: \omega \to \omega$ be the mapping assigning to a set its successor set:
- $n^+ := n \cup \left\{{n}\right\}$
Let $\varnothing \in \omega$ be the empty set.
Then $\left({\omega, \cdot^+, \varnothing}\right)$ is a Peano structure.
Proof
We need to check that all of Peano's axioms hold for $\left({\omega, \cdot^+, \varnothing}\right)$.
Suppose first that for $m, n \in \omega$, we have $m^+ = n^+$.
Since $n \in n^+$ it follows that $n \in m^+$.
Hence, either $n \in m$ or $n = m$.
Similarly, either $m \in n$ or $m = n$.
Now if $n \ne m$, both $m \in n$ and $n \in m$.
By Element of Minimal Infinite Successor Set is Transitive Set, it follows that $n \subseteq m$.
As $m \in n$, this contradicts Finite Ordinal is not Subset of one of its Elements.
Hence it must be that $n = m$, and Axiom $(P3)$ holds.
Next, since $n \in n^+$ for all $n \in \omega$, it follows that $n^+ \ne \varnothing$.
Hence, Axiom $(P4)$ holds as well.
Finally, let $S \subseteq \omega$ satisfy:
- $\varnothing \in S$
- $\forall n \in S: n^+ \in S$
Then by definition, $S$ is an infinite successor set.
Therefore, by definition of $\omega$ as the minimal infinite successor set:
- $\omega \subseteq S$
Consequently $S = \omega$ by the definition of set equality.
Thus Axiom $(P5)$ is seen to hold.
That is, $\left({\omega, \cdot^+, \varnothing}\right)$ is a Peano structure.
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 12$: The Peano Axioms
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.30$
- 2009: Steven G. Krantz: Discrete Mathematics Demystified: $\S 5.2$