Weekly Log:
Week 1
Over break and at the start of the semester I had been looking over “Lectures on Infinitary Model Theory” by David Marker. This book covers topics in infinitary logic. In particular, logics in which we are allowed both infinite conjunction and disjunction.
In a couple of early exercise’s, I was able to show that some nice properties from first order logic (namely Łoś theorem and Compactness) fail for these logics.
Compactness being the property that a collection of sentences has a model if and only if all finite subsets also have a model. Łoś being that property that given a ultra-filter and on a index I and a collection of models indexed by I, the corresponding ultra-product will satisfy a sentence if and only if the collection of indices of models which satisfy that sentence are in the ultra-filter. It is worth noting that Łoś implies Compactness. This can be shown by choosing the indexing set to be all finite subsets of a collection of sentences, then letting each model indexed by one of these subsets be one which satisfies that subset of sentences. It is then sufficient to show that this collection of subsets has an ultra-filter, and apply Łoś to show that the ultra-product satisfies all sentences in the collection.
It is pretty straight forward to show that these fail for infinitary logic’s, but it’s also not exactly surprising. See compactness says we only need to look at finite collections of sentences but in infinitary logic a single sentence can carry information about a infinite number of variables. Making it easy to construct a unsatisfiable collection of sentences in which each finite subset is satisfiable. Similarly, for Łoś, ultra-filters are only closed under finite intersections. In the proof of Łoś that closure is what translates to satisfying conjunctions of formulas, so it’s not shocking that allowing infinite conjunctions will cause problems.
The question then is what happens when we move the goalpost? Specifically, if we consider infinitary logic’s in which we are allowed countable conjunction and disjunction is there a version of compactness and Łoś which work for this logic? Dr. Boney suggested looking at if changing compactness to countable compactness (a set of sentences has a model iff every countable collection has a model) would work for this logic. I also considered looking into if regular compactness could work if we restrict the languages to only those which are finite. Dr. Boney also suggested considering whether I could modify the ultra-product definition to make an infinitary Logic version of Łoś.
Week 2
For this week I was able to show that Łoś can be recovered by simply changing the ultra-filter condition to be one which is closed under countable intersection instead of just finite. In general you could do the same for any cardinal.
This brings in a interesting question. Does countable Łoś imply countable compactness? See the reason Łoś implies compactness in first order is the ultra-filter lemma. This lemma states that any proper filter can be extend to an ultra-filter. In addition it is a proposition for filters that any subset of a powerset which has finite intersection property generates a proper filter. Both of these facts are important in showing that Łoś implies compactness in first order. The question then becomes, can any countable complete filter be extended to a countable complete ultra-filter? If so does each subset of a powerset with countable intersection property generate a countable complete filter?
Week 3
I was able to figure out in my meeting with Dr. Boney that there is not a countable version of the ultra-filter lemma. The ultra filter lemma requires that Zorn’s lemma holds on partial orders of filters with respect to inclusion. In other words every partial order chain in the set of filters needs a upper bound. In the usual proof this upper bound is provided by taking the union of all the filters in the chain. This works because the union is itself a filter. Seeing closure for superset is straight forward since any member of the union is also a member of a smaller filter. Closure under intersection can be shown by considering two elements call them f,g from the union. If f is in filter k in the chain and g is in filter h in the chain, then take the max of k and h call it m. Then f and g will both be in the filter corresponding to m and thus there is closer under finite intersection. The problem in applying this to countably complete filters is that we would need the union to be closed under countable intersection. This would require us to consider a countable collection of elements, which means there might not be a max index. Thus Zorn won’t hold.
Week 4
Had a busy week of school work but I did write an abstract for a talk. It will be on ultraproducts, namely what they are and hopefully if time permits a few key results. This is a challenging topic to cover as it incorporates several not obviously connected. In order to know what an ultraproduct is you need to have an understanding of models, ultrafilters, direct products and quotients. Each of these on their own can be challenging. To account for this I intend to lean more heavily on expressing the intuition for models and quotients, while more explicitly defining products and ultrafilters. The main goal is for the audience to understand the statement of Los and show a construction of the hyperreals using ultra-power. If time permits I would also like to walk though a proof of compactness.
Week 5
This week I read more of Maker’s book on infinitary model theory. The main ideas to come from this were fragments, back and forth arguments and Scott sentences. Some of these definitions where more challenging to parse so I also discussed them with Dr. Boney.
Fragments seem to be a key concept throughout the book so I want to make sure I understand them well. The problem with infinitary sentences as a general class is that we can lose some useful properties from first order logic such as compactness and Downward Lowenheim-Skolem. However, we still want to talk about theories which have natural axiomizations in infinitary logics. One way to due this is to consider collections of formulas in our infinitary language closed under nice first order properties. This is essentially what a fragment is.
Another useful concept in infinitary logics is that of a complete sentence. When allowing sentences to have arbitrary conjunction and disjunction it is possible to have a sentence that can satisfy and other infinitary or it’s negation. Interestingly a Scott sentence provides a explicit construction of such a sentence. It also provides an equivalence to back and forth arguments between models.
Week 6
This week I did a lot. I started chapter 3 of Marker’s book, which Dr. Boney helped me parse in our meeting. We also discussed what a model of topology would look like. Then I did my Talk on Ultraproducts.
The talk went over really well!! There was quite a few people who came. This talk was ambitious in that I needed to cover a lot of background before getting to the main topic. I did not want to fall into the common trap of sacrificing audience engagement for delivering more information. I made a effort to leave time for people form questions, and for me to make sure I gave satisfactory answers. I’m glad to say I believe I accomplished that. There was lots of good questions and I was able to answer all of them effectively. I was also still able to get to the main ideas at the end. I was pleased with the outcome and people seemed to enjoy it.
Chapter 3 is challenging to parse and may take me some time. In the meeting many things got clarified. The chapter is called ‘the space of countable models’. The idea is turn the collection of countable models in each countable vocabulary into a polish space. A polish space is a sort of tree like space. One example being cantor space. In fact if our vocabulary is just a relation then the space of models is homeomorphic to cantor space. Another useful property of these spaces is that the collection of models satisfying a given formula in countable infinitary logic is a borel subset of the space.
Making a model of topology turns out to be a funny exercise. Since a topology is a collection of subsets it would be most natural to use 2nd order logic. To talk about first order requires to form sorts. This is apparently still enough to talk about continues functions. However, Homotopy doesn’t play quite so nice with this. In fact it is a result from Peter Freyd that Homotopy isn’t even a concrete category (has a faithful functor into set).
Week 7
This week I was very ill and consequently behind on homework
Week 8
Spring break!!! Mostly caught up on homework though I did look at a bit more of chapter 3.
Week 9
This week we had the MAA sectional and talk by logic speaker Dr. Karen Lange. Her specialty is in computability theory. In her talk she talked about the computability of of certain binary trees. It was a lot of fun because it reminded me of a theory of automata class I took last semester. At lunch she also talked about how computability can be applied to model theory by seeing which theories have computable models.
I also spent more time on chapter 3 of the book. At this point I feel I understand the concept for the most part. It is mostly a contraction designed to look at the infinitary logic version of Vaught’s conjecture. Which is a conjecture on the number of countable models of a given theory. It was interesting to parse though I think I will be better served spending time on the next sections.