The Revised GCH – Some Motivation

One of the projects on my plate for next semester is to understand Shelah’s original proofs of his Revised GCH from SH460. Despite the fact that the proof from Abraham and Magidor’s chapter in the handbook is comparatively easy to work through, it looks like there’s some good information smuggled in Shelah’s original proofs which make them worth looking at. I should point out that yes, there are in fact two proofs of the Revised GCH in SH460. The first uses generic ultrapowers, and I’m a bit wary of it, as it uses Chapter 5 of Cardinal Arithmetic as a black box. The second proof, however is more pcf-theoretic, and it seems a bit less challenging since the aforementioned handbook chapter is such a wonderful resource on the basics of pcf theory.

Before I get to any of these proofs, I plan on actually working through the Abraham-Magidor version of the proof. I haven’t done any pcf theory for a few months, and I want to go back and get reacquainted with the machinery. Before even doing that though, I want to take some time to motivate why The Revised GCH is an appropriate name for the theorem. Hopefully this will have the benefit of getting other people interested in the result, because it is genuinely surprising and pretty. This part is intended for a mathematical audience acquainted with the basics of set theory. As a result, it will be simultaneously far too curt, contain far too much information, sweep too many details under the rug, and provide too many specifics.

In SH460, Shelah starts off with looking at Hilbert’s 1st problem: The continuum hypothesis. The question itself is rather simple: Is it the case that $2^{\aleph_0}=\aleph_1$? It turned out that this question was quite difficult to answer. In fact, this question itself spurred the development of quite a bit of set theory, but we won’t be focussing on that. What is worth noting is that we can generalize this question to the following:

“Is is the case that, for any cardinal $\kappa$, the operation $2^\kappa$ is precisely the cardinal successor operator on $\kappa$?”

A positive answer to the above question is called the General Continuum Hypothesis (or GCH). Godel was able to show that GCH was consistent with ZFC (the “usual” axioms of set theory that most mathematicians take). That is, we can find an example of a “universe” in which ZFC holds, and GCH is true.

Now, in order to answer this question further, we would need to know more about the map $\kappa \mapsto 2^\kappa$. However, for some time the only thing we knew was the fact that it must obey two rules:

1. If $\lambda\geq \kappa$, then $2^\lambda\geq 2^\kappa$;
2. $cf(2^\kappa)>\kappa$.

Due to the work of Easton, it turned out that these were the only rules for regular cardinals $\kappa$. That is, given any “function” $F$ from regular cardinals to cardinals obeying the two rules above, there is a universe in which the continuum function $\kappa\mapsto 2^\kappa$ is precisely described by $F$. This does provide a resolution to our question, but it seems very unsatisfying. Essentially, the continuum function on regular cardinals is arbitrary modulo two very minor restrictions. One thing to do here is to take this resolution as evidence that we’ve asked the wrong question, and instead look at how we can massage Hilbert’s problem into something more reasonable.

Here, we have two approaches. The first is to note that all of these issues are arising from the fact that we’re considering regular cardinals. So, we can ask ourselves about singular cardinals and see where we end up. This leads us to the singular cardinal hypothesis, and there has been a lot of fruitful investigation done in this vein by way of something called pcf theory. The other method is to see what we can say about regular cardinals, which is what we’re concerned about here.

One way of looking at GCH is that it says, roughly speaking “cardinal exponentiation is not too unruly”. So while we can’t say much about the continuum function, it may be worthwhile to look at the values of $\lambda^\kappa$ for $\kappa<\lambda$ regular. Perhaps we can ask that exponentiation behaves like sum and product for infinite cardinals, which brings us to the following first revision:

For regular cardinals $\kappa < \lambda$, we have $\lambda^\kappa=\lambda$

Still though, this is not quite what we want. Part of the issue is that these values are too tied up with each other, so failures for small values of $\kappa$ will imply failures all the way up. This is where Shelah introduces a revised version of cardinal exponentiation that allows for a finer slicing. First, some notation:

For cardinals $\kappa<\lambda$, we set $[\lambda]^\kappa=\{X\subseteq \lambda : |X|=\kappa\}$. We will also have occasion to use $[\lambda]^{<\kappa}=\bigcup_{\theta<\kappa} [\lambda]^\theta$.

One thing to note is that we have $|[\lambda]^\kappa|=\lambda^\kappa$, so looking at this collection isn’t completely unreasonable. For regular $\kappa<\lambda$, then we define “$\lambda$ to the revised power of $\kappa$” to be:

$\lambda^{[\kappa]}=\min\{|F| : F\subseteq [\lambda]^\kappa$ $\text{ such that for every }X\in[\lambda]^\kappa$ $\text{ there is some }G\subseteq F$ $\text{with }|G|<\kappa$ $\text{and} X\subseteq\bigcup G\}$

Now, this looks like a lot, but it’s not too bad. Essentially, we look at certain sorts of covering families $F\subseteq [\lambda]^\kappa$, and ask what the minimum cardinality of such a covering family must be. Here though, our version of covering is that any $X\in [\lambda]^\kappa$ is covered by a union of fewer than $\kappa$-many elements of $F$. The obvious question is “what does this have to do with $\lambda^\kappa$?”

Claim: For every $\lambda>\kappa$ we have that $\lambda^\kappa=\lambda$ if and only if $2^\kappa\leq \lambda$ and for every regular $\theta\leq\kappa$, $\lambda^{[\theta]}=\lambda$

First suppose that $\lambda^\kappa=\lambda$. Then certainly we have that $2^\kappa\leq\lambda$, as $2^\kappa\leq \lambda^\kappa$. Further, for any regular $\theta\leq \kappa$,  we know that

$|[\lambda]^\theta|=\lambda^\theta\leq\lambda^\kappa=\lambda.$

So we simply note that $F=[\lambda]^\theta$ witnesses that $\lambda^{[\theta]}\leq\lambda$. As the other inequality holds trivially, we’re done with this direction.

For the other direction, we proceed by induction on $\kappa$. So assume $\kappa=\aleph_0$,  and let $F=\{Y_i : i<\lambda\}$ be a family witnessing that $\lambda^{[\aleph_0]}=\lambda$. Let $X\in [\lambda]^{\aleph_0}$, and let $I\in [\lambda]^{<\aleph_0}$ be such that $X\subseteq \bigcup_{i\in I}Y_i$. Then $\bigcup_{i\in I} Y_i$ is countable, and so we see that $X$ is isomorphic to a subset $S$ of $\aleph_0$ insofar as it sits inside $\bigcup_{i\in I} Y_i$. Thus, we can associate to $X$ a unique $I\in [\lambda]^{<\aleph_0}$ and some $S\subseteq \aleph_0$, which yields an embedding of $[\lambda]^{\aleph_0}$ into $[\lambda]^{<\aleph_0}\times 2^{\aleph_0}$. By assumption, this set has size $\lambda$.

Now let $\kappa<\lambda$ be regular such that our conclusion holds for each $\theta<\kappa$. That is, for each such $\theta$, if $\lambda^{[\mu]}=\lambda$ for every regular $\mu\leq\theta$, then $\lambda^\theta=\lambda$. By assumption, we therefore know that $\lambda^\theta=\lambda$. Thus, we also have that $|[\lambda]^{<\kappa}|=\lambda$

As before, we begin by enumerating a family $F=\{ Y_i : i<\lambda\}$ witnessing the fact that $\lambda^{[\kappa]}=\lambda$. We then fix $X\in [\lambda]^\kappa$, and let $I\in [\lambda]^{<\kappa}$ be such that $X\subseteq \bigcup_{i\in I} Y_i$. As before, since $\bigcup_{i\in I}Y_i$ is of size $\kappa$, we see that $X$ is isomorphic to a subset $S$ of $\kappa$ insofar as it sits inside that union. So we can associate to $X$ a unique $I\in [\lambda]^{<\kappa}$ and some $S\subseteq \kappa$,which yields an embedding of $[\lambda]^\kappa$ into $[\lambda]^{<\kappa}\times 2^\kappa$. By assumption, this set has size $\lambda$, and so we’re done.

So that above claim shows that looking at these revised powers is  completely reasonable thing to do. The other nice thing is that Shelah and Gitik have shown that the values of $\lambda^{[\kappa]}$ and $\lambda^{[\mu]}$ are independent of each other for $\kappa<\mu<\lambda$. This brings us to the Revised GCH:

The Revised GCH Theorem (Shelah): Fix any uncountable strong limit cardinal $\mu$. For ever $\lambda\geq \mu$ there is some $\kappa<\mu$ such that if $\theta$ is regular with $\kappa\leq \theta<\mu$, then $\lambda^{[\theta]}=\lambda$.

Put more simply: for most pairs $(\lambda, \kappa)$, we have that $\lambda^[\kappa]=\lambda$. Given our discussion above, this is indeed a theorem deserving of the name “Revised GCH”.