I have now been embedded into the choice users for a week. The weekend provides a brief break I can use to safely gather my thoughts. Initially the choice users were very sensitive to my presence, apologizing to me whenever they blasphemed and applied the dreaded principle. By the end of the week however, they had already become desensitized to my presence. Choice is a hammer to them, and with it in hand, they see most problems as nails of various sizes. Blind persistence is the winning approach to these problems; if you keep hitting a nail it will eventually sink into the wood. And yet, we have discovered common ground. The shapeless, cyclopean morass generated from the power set operation is abhorrent to them as well. While I restrict my attention to the describable and the real to avoid this particular mathematical abomination, they use the tools of pcf and pseudo-power, which, when studied under the lens of choice, are house cats relative to the tiger that is the full power set. One, of course, should never underestimate a house cat.
On a serious note, the talks this week have been generally good, although 6 hours of new math day for a whole week is genuinely exhausting. Pretty much all of what we have been learning is the work of Shelah; the structure he created to be able to answer questions of singular cardinal combinatorics. James Cummings started the week and provided vital context and pictures that really help motivate and elucidate this somewhat dense material. Bill Chen also presented this week, and by all accounts, he covered a breathtaking amount of information. Thanks to him, we were able to see some of the major theorems of the initial setting of pcf theory (a singular cardinal with uncountable cofinality which is not a fixed point of the Aleph operation). James emphasized the topological connections of this material early on, and with this in mind I think much of what Bill proved can be seen as advanced topological theorems of ordinal spaces. In topology one can study the interplay between density and covering properties, and in pcf theory you can study the interplay between cofinality and covering properties. I think arguments can be made for cofinality being a reasonable replacement for density in ordinal spaces.
Although I am tired, I am looking forward to using this weekend to get some energy back and then jump right into more applications and developments of pcf theory next week.
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