# Uncountable ordinals without power set axiom

Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a set?

• That is a great question! – Asaf Karagila Sep 16 '10 at 13:12

No. One cannot prove that $\omega_1$ exists in ZF - P, with or without V = L.
The set $HC$ of hereditarily countable sets always satisfies ZF - P. (This is straightforward to check axiom per axiom.) Of course, $\omega_1 \notin HC$ but moreover every set in $HC$ is countable or finite as witnessed by a function in $HC$. Therefore, $HC$ knows that every set in $HC$ is countable and hence $HC$ is a model of ZF - P + "every set is at most countable."
Throwing in V = L into the mix doesn't help. Indeed, $HC^L = L_{\omega_1^L}$ (see my answer to your recent MO question for a proof) is a model of ZF - P + V = L + "every set is at most countable."
• Yes, Asaf. If $\kappa > \omega$ is regular then $H(\kappa)$, the family of sets whose transitive closure has size less than $\kappa$, is a model of ZFC-P in which all sets have size at most $\kappa$. (Regularity is needed for the replacement axiom to hold.) – François G. Dorais Sep 16 '10 at 16:18
• I only knew it was true for $H(\omega)$ for ZF-Inf. – Asaf Karagila Sep 16 '10 at 19:31