Natural Number is Transitive Set
Theorem
Let $n$ be a natural number.
Then $n$ is a transitive set.
Proof
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $n$ is a transitive set.
Basis for the Induction
From Empty Class is Transitive, we have that $\O$ is a transitive class.
By the Axiom of the Empty Set, $\O$ is a set.
Hence $\O$ is a transitive set.
Thus $\map P 0$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $k$ is a transitive set.
from which it is to be shown that:
- $k^+$ is a transitive set.
Induction Step
This is the induction step:
Let $k$ be a transitive set.
Then from Successor Set of Transitive Set is Transitive:
- $k^+$ is a transitive set.
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall n \in \N: n$ is a transitive set.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 3$ Derivation of the Peano postulates and other results: Theorem $3.1$