I have the following well-known exercise in cardinal arithmetics:

If $\kappa, \lambda$ are cardinals such that $\lambda$ is infinite, then $\kappa^{<\lambda}$ equals the supremum of the $\kappa^{\theta}$, where $\theta < \lambda$ is a cardinal.

1) Isn't this false for $\kappa=1$? The left side is $\lambda$ and the right side $1$. Remark that $\kappa^{<\lambda}$ is defined to be the cardinality of the disjoint(?) union of the function sets $map(\alpha,\kappa)$, where $\alpha < \lambda$ is an ordinal.

2) Is this supremum understood as a cardinal supremum or an ordinal supremum? If the latter is the case, why should the supremum be a cardinal at all?

3) Anyway, I don't know how to solve this exercise. I've already seen it several times, but 1) and 2) remain obscure in the literature I know.

  • $\begingroup$ Certainly, the sets $map(\alpha,\kappa)$ are disjoint for distinct $\alpha$: each element is a function with domain $\alpha$, so if $\alpha\neq\beta$, then $map(\alpha,\kappa)$ is disjoint from $map(\beta,\kappa)$ (the functions have different domains, so they are different as sets of ordered pairs). $\endgroup$ Commented Sep 1, 2010 at 19:10

1 Answer 1


First, (2): if you define a cardinal to be an ordinal that is not bijectable with any strictly smaller ordinal, then suppose that $\alpha$ is the ordinal supremum of $\kappa^{\theta}$, $\theta\lt\lambda$. Let $\beta$ be a strictly smaller ordinal than $\alpha$. By the definition of supremum, there must exist a $\theta\lt\lambda$ such that $\beta\lt\kappa^{\theta}\leq\alpha$; in particular, $\kappa^{\theta}$ cannot be bijected with $\beta$ (being a cardinal), and therefore neither can $\alpha$ (that would give an embedding of $\kappa^{\theta}$ into $\beta$, and Cantor-Bernstein would give you $|\beta|=\kappa^{\theta}$, contradicting that the latter is not bijectable with any strictly smaller ordinal). Thus, $\alpha$ is not bijectable with any strictly smaller ordinal, and so must be a cardinal. So whether you define the supremum as the ordinal-sup or the cardinal-sup, you will still get a cardinal (this holds for any set of cardinals).

Second, (1): you are correct that the definition does not match this for $\kappa=1$ (or for $\kappa = 0$); as noted by Carl in the comments, this is likely an erratum or ommission; it should hold for any $\kappa>1$.

Edit: The definition as a sum is done running over all ordinals, rather than all cardinals, so I'm fixing this below.

Finally, (3): you are trying to show that the supremum of the $\kappa^{\theta}$ equals the sum over all ordinals $\alpha<\lambda$ of $|\kappa^{\alpha}$, assuming $\kappa\geq 2$ (to prevent the problems noted). This because $|\kappa^{\alpha}|=|map(\theta,\kappa)|$ by definition, and the cardinality of the disjoint union is the cardinal sum. I believe this can be shown by transfinite induction on $\lambda$ as I do below, but there probably is a simpler method.

So, the proposition we want to show is that for any infinite ordinal $\lambda$, we have $$\sup_{\alpha\lt\lambda}|\kappa^{\alpha}| = \sum_{\alpha\lt\lambda}|\kappa^{\alpha}|.$$

First, the equality holds for $\lambda=\omega$: if $2\leq \kappa\lt\aleph_0$, then $\sup\{|\kappa^n|\,|\, n=0,1,2,3,\ldots\} = \aleph_0$ and $\sum_{n=0}^{\infty}|\kappa^n| = \aleph_0$; if $\aleph_0\leq\kappa$, then $|\kappa^n|=\kappa$ for all $n$, and $\sum_{n=0}^{\infty}|\kappa^n| = \sum_{n=0}^{\infty}\kappa = \kappa\aleph_0=\kappa$, so both sides agree.

Assume the result holds for $\lambda$; then the supremum of $|\kappa^{\alpha}|$ with $\alpha\lt\lambda^+$ is $|\kappa^{\lambda}|$; on the other hand, $$\sum_{\alpha\lt\lambda^+}|\kappa^{\alpha}| = \left(\sum_{\alpha\lt\lambda}|\kappa^{\alpha}|\right) + \kappa^{\lambda} = \sup_{\alpha\lt\lambda}|\kappa^{\alpha}|+|\kappa^{\lambda}|=|\kappa^{\lambda}|,$$ where the last equality holds because $|\kappa^{\alpha}|\leq |\kappa^{\lambda}|$ for each $\alpha\lt \lambda$, so the supremum is at most $|\kappa^{\lambda}|$, and the sum of two infinite cardinals is equal to their maximum. So again the two expressions agree.

Finally, we want to show that if $\lambda$ is a limit ordinal and the result holds for all $\beta\lt\lambda$, then it holds for $\lambda$. Then $$\sup_{\alpha\lt\lambda}(|\kappa^{\alpha}|) = \sup_{\beta\lt\lambda}\left(\sup_{\alpha\lt\beta}(|\kappa^{\alpha}|\right) = \sup_{\beta\lt\lambda}\sum_{\alpha\lt\beta}|\kappa^{\alpha}| = \sum_{\alpha\lt\lambda}(|\kappa^{\alpha}|).$$ So the equality holds for $\lambda$ as well. This establishes the result by transfinite induction for all infinite ordinals $\lambda$

Further Edit, 2 Sep 2010: Clarify how to finish it off.

So, the above shows that for any ordinal $\lambda$, $\sup_{\alpha\lt\lambda}|\kappa^{\alpha}| = \sum_{\alpha\lt\lambda}|\kappa^{\alpha}|$. To finish the exercise, we need to show that if $\lambda$ is a cardinal (that is, an ordinal that is not bijectable with any strictly smaller ordinal), then $\sup\{|\kappa^{\alpha}|\colon\alpha$ is an ordinal and $\alpha\lt\lambda\} = \sup\{|\kappa^{\theta}|\colon \theta$ is a cardinal and $\theta\lt\lambda\}$. To see this, note that $|\kappa^{\alpha}|=|\kappa|^{|\alpha|}$, and since $\lambda$ is assumed to be a cardinal, if $\alpha\lt\lambda$, then there exists a cardinal $\theta$, $\theta\lt\lambda$, such that $|\alpha|=|\theta|$, and hence $|\kappa^{\alpha}|=|\kappa^{\theta}|$. Thus, the two sets are equal, so their suprema are equal as well.

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    $\begingroup$ The question matches up with the definitions in Kunen's book, and the exercise mentioned above is problem 15 on page 44. Kunen defines $\kappa^{<\lambda}$ on p. 34. I think this may just be an erratum. $\endgroup$ Commented Sep 1, 2010 at 19:08
  • $\begingroup$ Thanks for the citation. I think the only problems arise if $\kappa = 0$ or $\kappa=1$; otherwise, I think the sum equals the supremum (as all summands will be infinite), so it is indeed likely to be just an error of omission. $\endgroup$ Commented Sep 1, 2010 at 19:27
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    $\begingroup$ Thanks. But this uses the definition $\kappa^{<\lambda} = \sum_{\theta} |\kappa^\alpha|$, where $\theta$ runs through the cardinals below $\lambda$. However, in Kunen's book, it runs through all ordinals. $\endgroup$ Commented Sep 2, 2010 at 0:00
  • $\begingroup$ This shouldn't be a problem with the induction. Let me try fix it and see. $\endgroup$ Commented Sep 2, 2010 at 2:22
  • $\begingroup$ Sorry, another correction: The exercise talks about the supremum of the $\kappa^{\theta}$, where $\theta$ are cardinals. However, in the sum, we take arbitrary ordinals. This is actually the whole problem ... $\endgroup$ Commented Sep 2, 2010 at 7:41

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