Does the definition of countable ordinals require the power set axiom?

Question

I am trying to understand the consequences of the different axioms of ZFC. In particular, I was trying to understand what you get on ZFC-power set (ZFC minus the power set axiom). If you have any references that I could read please let me know. In particular, I have a question. From the definition of ordinal number (for instance, Jech, p.19 and above) I believe that you do not need the power set axiom to define infinite ordinals beyond $\omega$, but I am not completely sure. If you don't need it, which is the largest ordinal that you can reach without using power set? can you reach $\omega_1$?

Answer 1

You may be very interested in the following paper:

Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone, What is the theory ZFC without power set? (arXiv, 2011)

Note that you cannot prove the existence of $\omega_1$ either, this is because $H(\omega_1)$, the set of the hereditarily countable sets contain all the countable ordinals, and it is a model of $\sf ZFC-Pwr$, but $\omega_1\notin H(\omega_1)$ because it is not hereditarily countable.