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It is well-known that in $\mathsf{ZF}$ there cannot exist a set containing all of the ordinals (transitive sets of transitive sets).

Is the same true for cardinals? (ordinals satisfying $(\forall \alpha\in\kappa)[\alpha\lnsim\kappa]$)

Of course the answer is "yes" assuming the axiom of choice because we can define successor cardinals, but what about a setting without choice?

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  • $\begingroup$ If $A$ is a set, the cardinality of $\mathcal P(\bigcup A)$ is strictly larger than the cardinality of any set in $A$. $\endgroup$ – Andrés E. Caicedo Nov 17 '18 at 16:17
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Here's a simple proof that the class of cardinals $(\dagger)$ (in $\mathrm{ZF}$) is a proper class:

We already now that the class of ordinals $\mathrm{Ord}$ is a proper class and as such unbounded in the von Neumann hierarchy. Now just note that for every $\alpha$ there is a cardinal $\alpha^+ > \alpha$ (this doesn't use any form of choice -- see Hartogs number). Hence the class of cardinals is unbounded in the von Neumann hierarchy (as it is $\in$-cofinal in $\mathrm{Ord}$) and as such it must be a proper class.

$(\dagger)$ This shows a bit more: The class of wellordered cardinals (or alephs) is a proper class.

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    $\begingroup$ Hartogs number! I notice that the axiom of replacement is crucial here because $V_{\omega+\omega}$ has very few cardinals. $\endgroup$ – Alberto Takase Nov 17 '18 at 10:07
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    $\begingroup$ @AlbertoTakase Replacement certainly plays a role here but notice that the powerset axiom is also important: Let $\kappa$ be in infinite cardinal. Then (assuming $\mathrm{ZFC}$ in our background universe) $(H_{\kappa+}; \in) \models \mathrm{ZFC}^-$, that is $\mathrm{ZFC}$ without the powerset axiom and $(H_{\kappa^+}; \in) \models \forall x \exists f \colon \kappa \to x \text{ surjective}$. In particular $(H_{\kappa^+}; \in) \models \kappa \text{ is the largest cardinal}$. $\endgroup$ – Stefan Mesken Nov 17 '18 at 10:16
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    $\begingroup$ @Alberto: Insisting that cardinals are ordinals is missing the point of "cardinals". $\endgroup$ – Asaf Karagila Nov 17 '18 at 18:22

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