Finite Ordinal is equal to Natural Number

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Theorem

Let $n$ be an element of the minimally inductive set.

Let $x$ be an ordinal.

Then:

$n \sim x \implies n = x$

Proof

Let $n \ne x$.

Then either $n < x$ or $x < n$ by Ordinal Membership is Trichotomy.

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If $x < n$, then by Subset of Finite Set is Finite both $x$ and $n$ are finite.

Therefore by No Bijection between Finite Set and Proper Subset:

$x \not \sim n$

Suppose $n < x$.

Then $x \sim n$ implies that $x$ is finite by definition.

By No Bijection between Finite Set and Proper Subset:

$x \not \sim n$

Therefore by Rule of Transposition:

$n \sim x \implies n = x$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.20$