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I am trying to fill in the details for part (ii) of Theorem 5.15 in Jech's Set Theory:

Theorem 5.15 If GCH holds and $\kappa$ and $\lambda$ are infinite cardinals, then

(i) If $\kappa \leq \lambda$, then $\kappa^\lambda = \lambda^+$.

(ii) If $\mathrm{cf}(\kappa) \leq \lambda < \kappa$, then $\kappa^\lambda = \kappa^+$.

(iii) If $\lambda < \mathrm{cf}(\kappa)$, then $\kappa^\lambda = \kappa$.

In the proof of (ii) he just states that it follows from these two lemmas:

Lemma 5.7 If $|A| = \kappa \geq \lambda$, then the set $[A]^\lambda$ has cardinality $\kappa^\lambda.$

(Here $[A]^\lambda = \{X \subset A : |X| = \lambda\}.$)

Lemma 5.8 If $\lambda$ is an infinite cardinal and $\kappa_i > 0$ for each $i < \lambda$, then

$$\sum_{i<\lambda}{\kappa_i} = \lambda\cdot\sup_{i<\lambda}{\kappa_i}.$$

I am struggling to construct an explicit proof using these results and a few facts about cardinal arithmetic such as absorption for infinite cardinals and cardinalities of power sets.

Since, assuming GCH, $\kappa^+ = 2^\kappa = |\mathcal{P}(A)|$ for some $A$ with cardinality $\kappa$, I thought I could come up with some sequence $\{\mu_i : i < \mathrm{cf}(\kappa)\}$ such that

$$\kappa^+ = 2^\kappa = |\mathcal{P}(A)| = \sum_{i < \mathrm{cf}(\kappa)} |[A]^{\mu_i}| = \sum_{i < \mathrm{cf}(\kappa)} \kappa^{\mu_i} = \mathrm{cf}(\kappa)\cdot\sup_{i<\mathrm{cf}(\kappa)}\kappa^{\mu_i} = \mathrm{cf}(\kappa)\cdot\kappa^\lambda = \kappa^\lambda$$

Is this approach a good idea? How would I go about finding the appropriate $\mu_i$? I suppose it will have to make use of the assumption that $\mathrm{cf}(\kappa) \leq \lambda < \kappa$, but I don't see how.

I'd much appreciate any hints on how to proceed (either with my suggestion or another way).

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You fell into a trap that also caught me when I was just starting to read mathematical books. When you see a reference to (5.7) it does not mean Lemma 5.7, but rather the displayed formula whose tag is 5.7, which in this case is the formula:

$$\kappa\leq\kappa^\lambda\leq 2^\kappa.$$ Similarly, (5.8) refers to the inequality $$\kappa<\kappa^\lambda\quad\text{ if }\lambda\geq\operatorname{cf}\kappa.$$

And indeed these are the inequalities needed to prove (i) and (ii).

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  • $\begingroup$ Oh! That makes a lot more of sense. Thanks! I think it was more confusing as the line immediately above cites Lemma 5.6, then it cites (5.7) and (5.8). $\endgroup$ – ryan221b Dec 4 '18 at 16:55

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