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.
Saturday, July 30, 2016
Monday, July 18, 2016
7/18/2016
Over the weekend, I started working on the introduction to my dissertation, and built the first draft of my cv. I've read through the first chapters of "Introduction to Cardinal Arithmetic" now in preparation for the summer school in set theory, which starts next week. I also begin reading Magidor's cardinal arithmetic chapter in the handbook today.
I believe we are really quite close to cracking the Harrington-Shelah paper. One more definition needs to be sorted out, and then the rest of details, while non-trivial, should be able to be filled in. It's frustrating that the original write-up is so incomplete, particularly since we need to generalize out from it. I still agree with Jackson that re-writing the arguments in terms of infinity Borel codes and admissable ordinals is the right approach, but the translation is not quite straightforward. It would be great if there was a source on this.
I believe we are really quite close to cracking the Harrington-Shelah paper. One more definition needs to be sorted out, and then the rest of details, while non-trivial, should be able to be filled in. It's frustrating that the original write-up is so incomplete, particularly since we need to generalize out from it. I still agree with Jackson that re-writing the arguments in terms of infinity Borel codes and admissable ordinals is the right approach, but the translation is not quite straightforward. It would be great if there was a source on this.
Friday, July 1, 2016
Some Thoughts on Size and Choice
My thesis (which is admittedly biased given my research topic) is that, while choice makes certain questions about combinatorics possible, it also covers up interesting problems regarding cardinalities.
The first of these distinctions that it obscures is the fundamental difference between surjections and injections. In the choice context, injections and surjections are perfectly dual: any covering map can be turned into embedding the other way, and any embedding can be turned into a covering map. Now even with the axoim of choice, this perfect duality breaks down once more structure is involved. Without choice you can still get surjections from injections, but divining injections from surjections is largely impossible.
Although this is a simple point, I think it is worth noting how structure is preserved differently by injections and surjections. In particular I want to focus on order structures, although I am sure there are interesting things to say with regards to other types of structures. Now if A maps onto B and has some order structure, nothing about this needs to pass onto B. Basically the fibers of the surjection can interact with the order in very messy ways. However, if A maps into B, then we know two things: that B is capable of containing an ordered substructure with the same order as A, and that an order structure can be put on A which satisfies at least the non-quantified properties of the order structure on B. For example, the power set of the reals maps onto the reals, but without extra universe assumptions, the power set of the reals cannot be linearly ordered. If you can inject a set A into the reals though, it can clearly be linearly ordered. A need not inherit all of the nice properties of the reals: for instance the rationals map into the reals but are not closed under supremums. Note also that just because the reals maps onto some set, it doesn't mean that set can even be linearly ordered. The example here would be any hyperfinite quotient of the reals.
One kind of structure which acts differently with respect to this distinction are well-orders. If an ordinal maps onto a set B, then we can pull representatives from the fibers of the corresponding surjection. So that ordinal would have to contain a perfect facsimile of B. In particular if B has a certain order structure, then that ordinal would have to permit an order structure which contains B's order structure as a sub-structure. On the other hand, it would also mean that B has to itself be well-orderable. In a universe with sets that cannot be well-ordered, this puts severe limitations on what ordinals can map onto. It is actually easier to map an ordinal into a set, as this only forces the set contain a copy of that ordinal. In particular the set can fail to permit a global well-order while admitting order structure which are "locally" well-orderable. An interesting example here is that, in L(R), many ordinals can map into the power set of the reals; as we mentioned before, however, this power set can not even by linearly ordered, let alone well-ordered.
Without the axiom of choice, the question "Which partial order structures does A admit?" is non-trivial. To ask "Is |A| < |B|?" then asks "For every partial order structure which A admits is there a partial order structure on B which contains that as a substructure?" I think this re-interpretation is the most intuitive way to "explain" why its not always possible to compare the cardinality of two sets.
This leaves me wondering how much size questions should depend on structure questions. The axiom of choice looks to minimize this distinction. On the other end of the spectrum, why should A inject into B unless we have explicitly built B on top of A? In this light, I suspect the field of invariant descriptive set theory is exploring what happens when this distinction is maximized (up to at least assuming countable choice). The framework of determinacy appears to be somewhere in between these two perspectives.
I have a final stray thought. Are different versions of the axiom of choice just different assertions about the lattice structure of the universe? If so, are there unexplored variants of the axiom of choice corresponding to different lattice structures? What kind of structure is compatible with ZF in general?
The first of these distinctions that it obscures is the fundamental difference between surjections and injections. In the choice context, injections and surjections are perfectly dual: any covering map can be turned into embedding the other way, and any embedding can be turned into a covering map. Now even with the axoim of choice, this perfect duality breaks down once more structure is involved. Without choice you can still get surjections from injections, but divining injections from surjections is largely impossible.
Although this is a simple point, I think it is worth noting how structure is preserved differently by injections and surjections. In particular I want to focus on order structures, although I am sure there are interesting things to say with regards to other types of structures. Now if A maps onto B and has some order structure, nothing about this needs to pass onto B. Basically the fibers of the surjection can interact with the order in very messy ways. However, if A maps into B, then we know two things: that B is capable of containing an ordered substructure with the same order as A, and that an order structure can be put on A which satisfies at least the non-quantified properties of the order structure on B. For example, the power set of the reals maps onto the reals, but without extra universe assumptions, the power set of the reals cannot be linearly ordered. If you can inject a set A into the reals though, it can clearly be linearly ordered. A need not inherit all of the nice properties of the reals: for instance the rationals map into the reals but are not closed under supremums. Note also that just because the reals maps onto some set, it doesn't mean that set can even be linearly ordered. The example here would be any hyperfinite quotient of the reals.
One kind of structure which acts differently with respect to this distinction are well-orders. If an ordinal maps onto a set B, then we can pull representatives from the fibers of the corresponding surjection. So that ordinal would have to contain a perfect facsimile of B. In particular if B has a certain order structure, then that ordinal would have to permit an order structure which contains B's order structure as a sub-structure. On the other hand, it would also mean that B has to itself be well-orderable. In a universe with sets that cannot be well-ordered, this puts severe limitations on what ordinals can map onto. It is actually easier to map an ordinal into a set, as this only forces the set contain a copy of that ordinal. In particular the set can fail to permit a global well-order while admitting order structure which are "locally" well-orderable. An interesting example here is that, in L(R), many ordinals can map into the power set of the reals; as we mentioned before, however, this power set can not even by linearly ordered, let alone well-ordered.
Without the axiom of choice, the question "Which partial order structures does A admit?" is non-trivial. To ask "Is |A| < |B|?" then asks "For every partial order structure which A admits is there a partial order structure on B which contains that as a substructure?" I think this re-interpretation is the most intuitive way to "explain" why its not always possible to compare the cardinality of two sets.
This leaves me wondering how much size questions should depend on structure questions. The axiom of choice looks to minimize this distinction. On the other end of the spectrum, why should A inject into B unless we have explicitly built B on top of A? In this light, I suspect the field of invariant descriptive set theory is exploring what happens when this distinction is maximized (up to at least assuming countable choice). The framework of determinacy appears to be somewhere in between these two perspectives.
I have a final stray thought. Are different versions of the axiom of choice just different assertions about the lattice structure of the universe? If so, are there unexplored variants of the axiom of choice corresponding to different lattice structures? What kind of structure is compatible with ZF in general?
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