(adsbygoogle = window.adsbygoogle || []).push({}); Which is bigger, a^b or b^a? (set theory)

Hi!

Thanks for letting me join your physics forums!

Will anyone help me with a set theory question I have? I've been racking my brains over this for the last two hours with no progress.

Which is greater using ordinal exponentation: [tex]\omega^{\omega_1}[/tex] or [tex]\omega_1^{\omega}[/tex]?

P.S. I know that [tex]\omega^{\omega_1}[/tex] equals the order type of [tex]\underbrace{ \omega \times \omega \times \omega \times ... }_{\omega_1 \ many \ times}[/tex], and [tex]\omega_1^{\omega}[/tex] equals the order type of [tex]\underbrace{ \omega_1 \times \omega_1 \times \omega_1 \times ... }_{\omega \ many \ times}[/tex], but I'm still stuck.

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# Which is bigger, a^b or b^a?

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