Existence of Ordinal with no Surjection from Set

Theorem

Let $S$ be a set.

Then, there exists a non-empty ordinal $\alpha$ such that there is no surjection from $S$ to $\alpha$.

Proof

By Hartogs' Lemma, let $\alpha$ be an ordinal such that there is no injection from $\alpha$ to $\powerset S$.

From Empty Mapping is Injective, it follows that $\alpha$ is non-empty.

Aiming for a contradiction, suppose there is a surjection $\phi : S \to \alpha$.

Define $\psi : \alpha \to \powerset S$ as:

$\map \psi \gamma = \set {x \in S : \map \phi x = \gamma}$

Let $\beta, \gamma \in \alpha$ such that:

$\map \psi \beta = \map \psi \gamma$

Since $\phi$ is a surjection, there is some $x \in S$ such that:

$\map \phi x = \beta$

from which it follows that:

$x \in \map \psi \beta = \map \psi \gamma$

But then, by definition:

$\map \phi x = \gamma$

so therefore, by definition of mapping:

$\beta = \gamma$

Thus, by definition, $\psi$ is an injection from $\alpha$ to $\powerset S$.

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