The last few posts I’ve been covering a characterisation of pointclasses that admit scales. To make scale theory even more confusing there’s a completely different notion of scale, which really has nothing to do with our previous one — this one being of a more combinatorial nature. To avoid unnecessary confusion I’ll call these new objects pfc scales (but usually they’re simply called scales as well, however).
The pcf scales are always associated to some singular cardinal , so let’s fix one of those, and also fix an associated increasing sequence of regular cardinals cofinal in . Then a pcf scale on with respect to is a sequence such that
- for every ;
- for every , meaning that for a tail of ;
- Given any there’s an such that (i.e. is a dominating family).
The following ZFC theorem of Shelah shows that we can always find pcf scales. For a proof of this, see e.g. Eisworth’s chapter in the handbook.
Theorem (Shelah). If is singular then there exists an increasing sequence of regular cardinals cofinal in with and a pcf scale on with respect to .
To exceed ZFC we strengthen the notion of pcf scale, and say that is a very good pcf scale on with respect to if it’s a pcf scale such that whenever satisfies then there’s a club and such that for every with , holds for every .
It turns out that the existence of very good pcf scales are related to the existence of square sequences, so let’s quickly recall what those are.
Definition. Let be cardinals. Then holds if there’s a sequence such that for each ,
- , and every is club in ;
- If then for every ;
- holds for every and .
Recall that the existence of square sequences can be seen as a “non-compactness property”, in that weak compactness implies failure of square principles, as we covered in a previous post. The following result shows that very good pcf scales are also signs of non-compactness.
Theorem (Cummings-Foreman-Magidor). Let be a singular cardinal and assume holds for some . Then there’s a very good pcf scale on .
Before we dig into the proof, we note that does not suffice for this theorem, as is shown in Levine (’15). It’s also shown in Gitik & Sharon (’08) that the existence of a very good pcf scale on doesn’t imply the weak square and in particular doesn’t imply either, making the existence of very good pcf scales and the existence of -sequences independent of each other (modulo large cardinals). Okay, let’s dig in.
Proof. Pick an increasing cofinal sequence in with . Let witness . Build recursively, where we inductively make sure that for all , with being the pcf scale on , given to us by Shelah’s theorem above.
For we simply let . For successors we again simply choose . Assume lastly that is a limit. We need to ensure that
- for every ;
- for every .
Points (1) and (2) are simple to achieve, and (c) can be achieved by using the regularity of . We now check that this actually works. So let satisfy and pick . Let be such that , let with , and let .
Then by (3) of the definition of , and . Also, and . We chose in the above condition (3). QED