Using Replacement Schema we can prove any well-ordering is isomorphic to an ordinal number.
Q: Is the following consistent?
$ZFC-Rep+\neg Rep+\text{Any well-ordering is isomorphic to an ordinal number}$
Using Replacement Schema we can prove any well-ordering is isomorphic to an ordinal number.
Q: Is the following consistent?
$ZFC-Rep+\neg Rep+\text{Any well-ordering is isomorphic to an ordinal number}$
Let $\kappa$ be a $\beth$-fixed point -- that is, let $\kappa = \beth_\kappa$. Then $V_\kappa$ models ZFC - Rep + "every well-order is isomorphic to an ordinal". But ZFC proves the existence of $\beth$-fixed points. Thus, when $\kappa$ is the least such fixed point, $V_\kappa$ models ZFC - Rep + "every well-order is isomorphic to an ordinal" but fails to model ZFC (since being a $\beth$-fixed point is absolute for $V_\kappa$).