The continuum hypothesis CH is the assertion that the size of the power set of a countably infinite set $\aleph_0$ is the next larger cardinal $\aleph_1$, or in other words, that $2^{\aleph_0}=\aleph_1$. The generalized continuum hypothesis GCH makes this same assertion about all infinite cardinals, namely, that the power set of any infinite cardinal $\kappa$ is the successor cardinal $\kappa^+$, or in other words, $2^\kappa=\kappa^+$.

Yesterday I received an email from Geoffrey Caveney, who proposed to me the following axiom, which I have given a name.

The** Alternative GCH** is the assertion that for every cardinal number $\kappa$, there is a set $F$ of cardinals such that $P_F(\kappa)=\{X\subseteq\kappa\mid |X|\in F\}$ has size $\kappa^+$.

The set $P_F(\kappa)$ restricts the power set operation to consist only of subsets whose size is allowed by the family $F$. We may assume without loss that $F$ consists of cardinals of size $\kappa$ or less.

Caveney was excited about his axiom for three reasons. First, a big part of his motivation for considering the axiom was the observation that the equation $2^\kappa=\kappa^+$ is simply not correct for finite cardinals $\kappa$ (other than $0$ and $1$) — and this is why the GCH makes the assertion only for infinite cardinals $\kappa$ — whereas the alternative GCH axiom makes a uniform statement for all cardinals, and gets the right answer for the finite cardinals. Specifically, for any natural number $n$, we can let $F=\{0,1\}$, and then note that $n$ has exactly $n+1$ many subsets of size in $F$. Second, Caveney had also observed that the GCH implies his axiom, since as we just mentioned, it is true for the finite cardinals and for infinite $\kappa$ we can take $F=\{\kappa\}$, using the fact that every infinite cardinal $\kappa$ has $2^\kappa$ many subsets of size $\kappa$ (we are working in ZFC). Third, Caveney had noticed that his axiom implies the continuum hypothesis, since in the case that $\kappa=\aleph_0$, there would be a family $F$ for which $P_F(\aleph_0)$ has size $\aleph_1$. But since there are only countably many finite subsets of $\aleph_0$, it follows that $F$ must include $\aleph_0$ itself, and so this would mean that $\aleph_0$ has only $\aleph_1$ many infinite subsets, and this implies CH.

To my way of thinking, the natural question to consider was whether Caveney’s axiom was actually weaker than GCH or not. At first I noticed that the axiom implies $2^{\aleph_1}=\aleph_2$ and similarly $2^{\aleph_n}=\aleph_{n+1}$, getting us up to $\aleph_\omega$. Then, after a bit I noticed that we can push the argument through all the way.

**Theorem**. The alternative GCH is equivalent to the GCH.

**Proof**. We’ve already argued for the converse implication, so it remains only to show that the alternative GCH implies the GCH. Assume that the alternative GCH holds.

We prove the GCH by transfinite induction. For the anchor case, we’ve shown already above that the GCH holds at $\aleph_0$, that is, that CH holds. For the successor case, assume that the GCH holds at some $\delta$, so that $2^\delta=\delta^+$, and consider the case $\kappa=\delta^+$. By the alternative GCH, there is a family $F$ of cardinals such that $P_F(\kappa)=\kappa^+$. If every cardinal in $F$ is less than $\kappa$, then $P_F(\kappa)$ has size at most $\kappa^{<\kappa}=(\delta^+)^\delta=2^\delta=\delta^+=\kappa$, which is too small. So $\kappa$ itself must be in $F$, and from this it follows that $\kappa$ has at most $\kappa^+$ many subsets of size $\kappa$, which implies $2^\kappa=\kappa^+$. So the GCH holds at $\kappa$, and we’ve handled the successor case. For the limit case, suppose that $\kappa$ is a limit cardinal and the GCH holds below $\kappa$. So $\kappa$ is a strong limit cardinal. By the alternative GCH, there is a family $F$ of cardinals for which $P_F(\kappa)=\kappa^+$. It cannot be that all cardinals in $F$ are less than the cofinality of $\kappa$, since in this case all the subsets of $\kappa$ in $P_F(\kappa)$ would be bounded in $\kappa$, and so it would have size at most $\kappa$, since $\kappa$ is a strong limit. So there must be a cardinal $\mu$ in $F$ with $\newcommand\cof{\text{cof}}\cof(\kappa)\leq\mu\leq\kappa$. But in this case, it follows that $\kappa^\mu=\kappa^+$, and this implies $\kappa^{\cof(\kappa)}=\kappa^+$, since by König’s theorem it is always at least $\kappa^+$, and it cannot be bigger if $\kappa^\mu=\kappa^+$. Finally, since $\kappa$ is a strong limit cardinal, it follows easily that $2^\kappa=\kappa^{\cof(\kappa)}$, since every subset of $\kappa$ is determined by it’s initial segments, and hence by a $\cof(\kappa)$-sequence of bounded subsets of $\kappa$, of which there are only $\kappa$ many. So we have established that $2^\kappa=\kappa^+$ in the limit case, completing the induction. So we get all instances of the GCH.

**QED**