Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Under what assumptions on an infinite cardinal $\kappa$ we have $$\kappa^\kappa= 2^\kappa?$$

Please delete this question. I know the answer.

share|cite|improve this question
Once posting a question, it might be useful for future visitors. If you know the answer it is fine to post an answer on your own. – Asaf Karagila Apr 21 '12 at 15:46
I'm pretty sure this is a repeat, and I know I've posted an answer. In fact, if $\lambda$ satisfies $2\leq \lambda \leq 2^{\kappa}$, then $\lambda^{\kappa}=2^{\kappa}$ by the same argument Asaf uses below. – Arturo Magidin Apr 21 '12 at 22:07
up vote 12 down vote accepted

Assuming $\kappa$ is an ordinal the answer is always.

The reason is simple: by Cantor's theorem we have $2<\kappa<2^\kappa$, therefore using exponentiation laws: $$2^\kappa\le\kappa^\kappa\le\left(2^\kappa\right)^\kappa=2^{\kappa\cdot\kappa}=2^\kappa$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.