# A question about a proof of Hausdorff's Formula

My textbook Introduction to Set Theory by Hrbacek and Jech presents Hausdorff's Formula:

and its corresponding proof:

I am unable to deduce 3. from 1. and 2. as stated in the proof.

1. Each function $$f:\omega_\beta \to \omega_{\alpha+1}$$ is bounded

2. The definition of ordinal exponentiation: $$\omega_{\alpha+1}^{\omega_\beta}=\sup \{\omega_{\alpha+1}^{\lambda} \mid \lambda< \omega_\beta\}=\bigcup_{\lambda< \omega_\beta}\omega_{\alpha+1}^{\lambda}$$

3. $$\omega_{\alpha+1}^{\omega_\beta}=\bigcup_{\gamma < \omega_{\alpha+1}}\gamma^{\omega_\beta}$$

Could you please elaborate on this point? Thank you for your help!

It is not ordinal exponentiation, it is cardinal exponentiation, i.e. $$\omega_{\alpha+1}^{\omega_\beta}$$ is the set of all functions $$\omega_\beta\to \omega_{\alpha+1}.$$ If every such function is bounded, then every such function is a function $$\omega_\beta\to \gamma$$ for some $$\gamma < \omega_{\alpha}$$ and hence $$\omega_{\alpha+1}^{\omega_\beta} = \bigcup_{\gamma < \omega_{\alpha+1}}\gamma^{\omega_\beta}.$$
• Thank you so much for your answer! I am now clear. To avoid this confusion, it is better to denote the set of all functions $f:X \to Y$ by $^X Y$ rather than $Y^X$. Although this denotation is very uncommon, I have seen some users doing so. – Le Anh Dung Dec 30 '18 at 4:18