Abstract: *We will talk about some cardinal characteristics at singular cardinals and see how they are both easier and more difficult than cardinal characteristics at regular cardinals.*

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GOST Seminar

Room 620, Hill Center

Department of Mathematics

110 Frelinghuysen Road

Piscataway, NJ 08854

GOSTS did not occur much of the semester except through occasional, more spontaneous meetings due to a variety of factors including a strike. As a result, most details about these meetings were not recorded.

Abstract: *We will talk about some cardinal characteristics at singular cardinals and see how they are both easier and more difficult than cardinal characteristics at regular cardinals.*

Abstract: *We go over the second section in the handbook chapter on Fine Structure: The First Projectum. This is one of the more fundamental concepts in fine structure, and is a bit more tangible, modulo some facts about \(\mathrm{L}\). For example, we can calculate the first projectum in many cases. Mostly this section is set up for understanding reducts and fine-structure preserving embeddings later.*

Abstract: * We will talk about some large cardinals between supercompact and \(0=1\), and their implications for arithmetic*

Abstract: * We go over the first section in the handbook chapter on Fine Structure: Acceptable J-Structures. Given that a lot of the introductory material on rudimentary functions was already considered earlier, much of this detail will be skipped in favor of filling out some details left missing in regards to acceptable structures, Q-formulas, cofinal maps, and so forth.*

Abstract: * With a small amount of background on soundness, a proof (sketch) of \(\square_\kappa\) in \(\mathrm{L}\) is finished.*

Brian talks about the wreath product and some analytic equivalence relations.

Abstract: * A proof (sketch) of \(\square_\kappa\) in \(\mathrm{L}\) (for every infinite \(\kappa\)) from fine structure is given as motivation to for the subject, hopefully more accessible than things related to core models and iteration trees, which themselves require a large amount of background before they can be seen as “useful” in other areas of set theory. Most of the details are left as mysterious black boxes.*

Abstract: * We are going to do what every calc student has always dreamed about-- plugging in infinity! More precisely we are going to study iterated real formal Laurent series with generalized exponents, which turn out to be quite useful for solving problems from classical calculus. This is an exposition of "Logarithmic-Exponential Power Series" by van den Dries, Macintyre and Marker*

Abstract: * We will talk about Woodin's extender algebra, which is one of the main tools in inner model theory.*

* The main reference is I. Farah, "The extender algebra and \(\Sigma^2_1\)-absoluteness."*

Brian discusses proofs of the consistency of the failure of choice, and introduces several questions we discuss.

Abstract: *The basics of extenders are introduced, and motivated. We proceed slowly as the definitions are complicated and usually aren’t given sufficient time to be digested. We begin with extenders derived from elementary embeddings, and then hopefully can give some ideas to motivate certain large cardinals.*

Abstract: *We discuss some history of cardinal arithmetic, prove Silver's theorem that GCH cannot fail for the first time at a singular cardinal with uncountable cofinality, and then briefly introduce Shelah's pcf (possible cofinality) theory. We will also show yet another reason 4 is the largest number.*

Abstract: *We will cover proofs of some of the fundamental properties of measurable cardinals, and introduce iterated ultrapowers and their basic properties. This will be the last talk in the series before introducing extenders.*

Abstract: *In honor of Halloween, I will talk about the scariest object: the set of all sets. Specifically, we'll motivate NF set theory, where there is a set of all sets and it's fine because you just don't ask bad questions. I will also show yet another reason 4 is the largest number.*

Abstract: *We will talk about the basics of elementary embeddings, and how this connects with the concept of an ultrapower mentioned at the end of the previous talk. Embeddings are technically higher-order objects, but ultrafilters allow us to capture them (or parts of them) in a first-order way. So frequently we are able to define large cardinals in a first-order way that ordinarily are defined by embeddings. Ultrapowers given by ultrafilters (over a cardinal) have limitations, but we must build intuition before we delve too deeply into overcoming these obstacles with extenders.*

Notes: [Measures and Ultrapowers] with more info to be found in [Some Notes on Set Theory].

Abstract: *We discuss some elementary facts about infinite sets, including how to compare their size, Cantor's diagonal method, Schroder-Bernstein theorem, Konig's theorem, etc. Time permitting we will also mention the inductive formula for cardinal exponentiation.*

Abstract: *Filters appear in a variety of places in mathematics, but it’s model theory where they can be used to generate new structures from old ones such that things that were “mostly” true are now just true. Applied to set theory, we get a nice theory of embeddings that marks the starting point to studying a lot of the larger large cardinal hierarchy. This study eventually progresses into extenders (a kind of “hyper measure”). In this talk, I will introduce filters, ultrafilters, and hopefully ultrapowers as well.*

Notes: [Measures and Ultrapowers] with some corrections, with more info to be found in [Some Notes on Set Theory].

Abstract: *I will finish talking about the motivation and terminology of forcing and then start talking about some examples and the kinds of arguments commonly encountered and glazed over.*

Notes: [The Theory of Forcing], with more info to be found in [Some Notes on Set Theory].

Abstract: *In this talk, I will go over all the basic facts about ordinals every set theorist should know. Then we'll use those facts to prove Zorn's lemma.*

Abstract: *The technique of forcing is usually the gateway to studying advanced set theory. The technique involves a lot of technical terminology that can be difficult to grasp without proper motivation, making further study more difficult. I will go over the motivation and terminology of forcing. This is mostly based off of a previous talk given at GOSTS in 2020.*

Notes: [The Theory of Forcing], with more info to be found in [Some Notes on Set Theory].

Abstract: *Introduced by J. D. Hamkins, gap forcing isn't so much a forcing notion as a property of forcing notions, and a very useful one at that. The main idea is the non-introduction of large cardinals with large forcings. Such results are known for small forcings of course, but the generalization is very useful especially to long Easton support iterations. In this talk, I'll go over the relevant concepts and results as proven in Hamkins' original paper "Gap Forcing".*

Brian finishes the seminar topic.

Brian continues the seminar topic.

Abstract: *Today we will examine models of the form \(\mathrm{HOD}(M\cup X)\). Using results from last time, we can show they all look like \(M[x]\) (meaning "the smallest model containing \(M\) and \(x\)"; not a forcing extension). We'll need this next week to describe symmetric models.*

Abstract: *In this talk, we'll finish showing the action of \(\mathrm{Aut}(B)\) on generics is transitive. Then we'll use that to give a purely combinatorial description of the boolean value \(\|x \in OD(M,y)\|\).*

Abstract: *As we've seen earlier this year, there are a few ways to build models where AC fails. Classically, this was done using a group action on the forcing poset to define sufficiently symmetric names. Another approach was to look at \(\mathrm{HOD}\) sets in a sufficiently interesting model. It turns out these approaches are actually the same; that is, every symmetric extension between \(M\) and \(M[G]\) has the form \(\mathrm{HOD}(M[x])\) for some \(x\) in \(M[G]\), and every model \(\mathrm{HOD}(M[x])\) is the symmetric model with respect to some group action on the filter. Our goal in these talks is to prove this interesting theorem of Grigorieff.*

*For the first talk, we will look closer at \(\mathrm{Aut}(P)\) when \(P\) is a boolean algebra for forcing, and we'll try to describe \(\|x\in OD(M,y)\|\) explicitly.*

GOSTS did not occur on 2021-03-17 due to spring break, nor on 2020-03-10 because we didn't feel like it.

Abstract: *In the standard consistency proof of the existence of a precipitous ideal, the precipitous ideal is induced by a generic elementary embedding, which is obtained by a lift up argument. To show its precipitousness, the calculation of its quotient algebra is crucial.*

*This argument, actually, can be regarded as a very special case of Foreman’s duality theorem.*

*In this talk, I will explain what Foreman’s duality theorem says and how to use it. For example, the preservation theorem of Kakuda and Baumgartner-Taylor will be shown as its corollaries.*

Abstract: *In this talk, I'll sketch a proof that \(\mathrm{Sh}(\mathbb{P})\) is a topos. I'll try to describe the internal logic of \(\mathbb{P}\), and we'll see that it think it satisfies a large fragment of ZFC (stated in it's internal language). I hope this will give a nice view of why \(\mathrm{ro}(\mathbb{P})\) comes up so naturally. If there's time, I'll also try to define a way due to Mitchell of constructing a real model of ZFC inside a topos.*

Abstract: *In this talk, I'll define a sheaf on a poset (takes most of the time), and how the topos of sheaves on \(\mathbb{P}\) is related to the class of \(\mathbb{P}\)-names. This is useful to know about if you ever want to convince algebraic geometry people that your work about forcing is somehow relevant to them.*

Abstract: *Prewellorders are a result of maps into ordinals. If we examine such maps with definability restrictions, we get a rich theory about these in relation to large cardinal and determinacy assumptions. These bring in new concepts like scales and various properties of pointclasses. Our main goal will be to introduce these topics enough to show that the perfect set property on \(\boldsymbol{\Pi}^1_1\) implies that \(\omega_1\) is inaccessible to the reals, the converse to what was proven in the previous talk in the series.*

Notes: Slides used, adapted from [Some Notes on Set Theory].

Abstract: *Prewellorders are a result of maps into ordinals. If we examine such maps with definability restrictions, we get a rich theory about these in relation to large cardinal and determinacy assumptions. These bring in new concepts like scales and various properties of pointclasses. Our main goal will be to introduce these topics enough to show that the perfect set property on \(\boldsymbol{\Pi}^1_1\) implies that \(\omega_1\) is inaccessible to the reals, the converse to what was proven in the previous talk in the series.*

Notes: Slides used.

Abstract: *Forcing the failure of AC via Cohen forcing is a pretty well-known theorem that involves a lot of new concepts worthy of discussion. In particular, I will discuss product forcing, (weak) homogeneity, and how these relate to common forcing notions. Then I will show how all of this comes together in forcing the failure of AC (in an inner model).*

Notes: Slides used, adapted from [Some Notes on Set Theory].

Abstract: *In the second session, I introduce Hamkins and Lewis' notions of "writable ordinals" (codes for ordinals that can be written by an ITTM program with input 0) and "clockable ordinals" (the possible run-times of an ITTM program with input 0). I prove various theorems about writable and clockable ordinals, including the supremum of the writable ordinals is admissible and the limit of admissibles, and that no clockable ordinal is admissible. I conclude with Philip Welch's proof that the suprememum of the writable ordinals is the supremum of the clockable ordinals.*

Abstract: *In this series of talks, I will introduce a model of computation proposed by Joel Hamkins and Andy Lewis. Infinite time Turing machines (ITTMs) are an analog of Turing machines that can run for transfinitely many stages. In the first session, I will describe how ITTMs operate and discuss which types of sets of integers and pointclasses that ITTMs can decide.*

Abstract: *A short introduction to the basic consequences of determinacy is given. This includes the usual three basic properties of sets of reals: the perfect set property, the Baire property, and Lebesgue measurability.*

Notes: Slides used.

Navin finishes the the seminar topic

Abstract: *Recall permutation models which are models of ZFA, set theory with atoms. We can make permutation models which some weak forms of the Axiom of Choice fail while the other hold. And, we were able to transfer those independent results to ZF. However, not all of such results can be transfer to ZF.*

Abstract: *The Lévy hierarchy of formulas gives us absoluteness for \(\Delta_1\)-definable relations. Going beyond this is not possible in general, but the definability results of the lightface hierarchies suggest another direction to go: absoluteness of analytical relations. In this talk, we will show Shoenfield's theorem of \(\Sigma^1_2\)-absoluteness, and then see how this relates to \(\mathrm{L}\) and results like \(\Sigma^1_1\)-boundedness.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *The lightface versions of the boldface hierarchies are introduced with a discussion of computability over polish spaces. We also relate these hierarchies to sets of reals definable over arithmetic and over (fragments of) set theory; connecting description, computability, and topology.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *We investigate three interesting properties sets of reals can have and think about what sets have these properties through the lens of the boldface hierarchies. In particular, we think about the concepts of the perfect set property, involving lots of set theory; the Baire property, involving lots of topology; and Lebesgue measurability, involving analysis.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *Descriptive set theory is a broad field, but all parts of it talk about and attempt to categorize how complex sets of reals are. What this means practically speaking is looking at the Borel and Projective hierarchies. Typically, rather than looking at \(\mathbb{R}\) directly, we study Polish spaces like Baire or Cantor space. In this talk, the basic theory of Polish spaces and their Borel and Projective hierarchies are introduced.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Brian finishes the seminar topic.

Brian continues the seminar topic.

Abstract: *The independence of choice from ZF was open for a long time, since it requires forcing. However, most of the actual work was done way back in the 1920s. The idea is in ZFA set theory (ZF with a set of non-empty atoms with no elements), a group can act on the atoms, and thus act on the whole universe (on the other hand, ZF models can't have automorphisms).*

* In this first talk I'll go through the machinery used to build a few permutation models of ZFA where AC fails (to varying degrees). Then we'll talk about how, by having a group act on a forcing poset, you can start to do this in ZF. The general theorem I'd like to prove is that any (rank initial segment of a) ZFA model is \(\in\)-isomorphic to a (rank initial segment of a) ZF model, which greatly simplifies a lot of arguments. We'll see if I get there Wednesday; it may be early next week.*

*In subsequent talks, I'd like to work towards the surprising theorem that it is consistent "choice from families of triples" is true, while "choice from families of pairs" is false.*

Abstract: *Martin's Axiom is an important axiom in set theory because of its great number of consequences and ease of use. One practical use of iterated forcing is proving the consistency of MA (without CH). This talk covers the proof, also introducing some general theory along the way.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *We continue the discussion of support, finishing the topic of the direct limit and introducing the inverse limit. Then if there's time, the idea of factoring is introduced.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *With two-step iterations defined, we can look at longer iterations and various ways of dealing with the limit stage in terms of support. We can then think about what properties these limit iterations have based on the (names for) preorders we force with and the support we allow.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *Results using forcing in set theory are often easier to phrase using iterated forcing. In this series, I will explain and motivate the very technical definitions of iterated forcing. The first talk in this series will focus on two-step iterations, looking at basic definitions, preservation properties, and canonical names.*

Notes: Slides used, adapted from [Some Notes on Set Theory]

Abstract: *We'll finally prove the comparison lemma, which is the most important basic result in inner model theory.*

Abstract: *I'll go over random forcing. This means reminding you how boolean valued models work with forcing, and a bunch of measure theory facts you'd forgotten. Then we'll iterate and construct a model where \(\mathrm{cov}(\mathcal{N})\) is \(\aleph_\omega\).*

Abstract: *We'll cover the content that we didn't have time to explain the last time, i.e., a normal iteration game.*

Abstract: *Cichoń's diagram is a partial order describing the ZFC-provable relationships between 10 cardinal invariants of the continuum, relating null sets, comeager sets, and the bounding and dominating numbers. A classical set theory problem, completed in 1999, was to prove that Cichoń's diagram completely describes these cardinals; i.e., for any not-obviously-inconsistent assignment of invariants to \(\aleph_1\) and \(\aleph_2\), constructing a model with those as the cardinaliites.*

*In this talk, we will prove the relationships in Cichoń's diagram. If time permits, we'll also start describing some forcings used to show the inequalities can be strict; although that will be our main topic next time.
*

Abstract: *We introduce a fine structural version of ultrapowers and iteration trees. Iteration trees should be regarded as a result of an iteration game, where two players will choose extenders and cofinal branches. A winning strategy for the player who chooses branches in this game is called an iteration strategy. Then we finally reach the definition of a mouse.*

GOSTS did not occur on 2020-03-18 due to spring break, nor on 2020-03-25 due to university operating status. From this point up, the seminar is held online.

Abstract: *With basic terminology out of the way, I will go through some simple examples of forcing posets, and the consequences of forcing with them. The hope is to give not only an intuition with forcing arguments, but also an intuition for why certain posets look the way they do.*

Notes: [The Theory of Forcing].

Abstract: *Standard fine structural notions will be defined for (coded) potential premice. Our definition is more or less the same as Mitchell–Steel's book, but we adopt Schlutzenberg's simplification. If time permits, I will explain the big picture of Mitchell–Steel's theory.*

Abstract: *I will go over the basic terminology and theory of forcing. Forcing is a fundamental technique in set theory mostly used for establishing consistency results. The technique, however, involves a lot of technical definitions and is often a student's first foray into more advanced set theory. So I hope to give some perspective into these definitions, showing why they make sense and are natural to consider.*

Notes: [The Theory of Forcing].

Abstract: *After going over the definition of a potential premouse (ppm), the amenable code of a ppm will be introduced to develop fine structure theory nicely.*

Abstract: *Ultrafilters can be seen in a variety of places around mathematics, especially within set theory. For our purposes, ultrafilters give rise to ultraproducts, and these have a nice relationship with elementary embeddings. This goes over the basics of what exactly these are and some of the common techniques used when studying them.*

Notes: [Measures and Ultrapowers].

Abstract: *A mouse is the central notion in inner model theory. I'll try to give a fully-detailed explanation of the basic theory of mice in Mitchell–Steel style. I'll also point out many errors in old literature.*

Abstract: *Often used without thinking, skolem hulls and their transitive collapses are basic tools in a set theorist's toolkit. I will go over the basic ideas, and applications (mostly just with \(\mathrm{L}\)) if time permits.*

Notes: [The Basics of Skolem Hulls and Transitive/Mostowski Collapses], which is more-or-less just a selection from [Some Notes on Set Theory].