Unique Ordinal Exponentiation Inequality
Theorem
Let $x$ and $y$ be ordinals.
Let $x > 1$ and $y > 0$.
Then there exists a unique ordinal $z$ such that:
- $x^z \le y$ and $y < x^{z^+}$
Proof
Existence of $z$
By Lower Bound for Ordinal Exponentiation, $y \le x^y$ so $y < x^{y^+}$.
Therefore, $y$ is bounded above by $x^w$ for some $w$, so there is a smallest $w$ such that:
- $y < x^w$ by Subset of Ordinals has Minimal Element.
Moreover, if $w$ is a limit ordinal then there is some $v ∈ w$ such that $y < x^v$, contradicting the fact that $w$ is minimal.
Therefore, $w$ is not a limit ordinal and therefore must be the successor of some ordinal.
So $w = z^+$ for some $z$.
Since $z < z^+$, it follows that $x^z ≤ y$ and $y < x^{z^+}$.
$\Box$
Uniqueness of $z$
Assume that $x^z \le y$ and $y < x^{z^+}$ for some $z = z_1$ and $z = z_2$.
Then $x^{z_1} < x^{z_2^+}$ and $x^{z_2} < x^{z_1^+}$.
By Membership is Left Compatible with Ordinal Exponentiation $z_1 < z_2^+$ and $z_1 \le z_2$.
Similarly $z_2 \le z_1$.
By definition of set equality:
- $z_1 = z_2$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.38$