# James Holland

## jch258@math.rutgers.edu

Main Page Class Material Resources GOST Seminar Schedule About Me
Room 618, Hill Center
Department of Mathematics
Piscataway, NJ 08854

### The Graduate Operated Set Theory (GOST) Seminar

#### 12:00 PM to 1:00 PM on Wednesdays

Until university operating status returns to normal, GOSTS will be held online via Canvas. Email me to recieve an invitation to the course page.

The seminar will run from 2020-09-09 to 2020-12-09.

#### 2020 (Fall) Schedule of Speakers

 2020-10-21 Brian Pinsky "ZFA and the Independence of Choice-like Principles" [Part 3] 2020-10-13 Brian Pinsky "ZFA and the Independence of Choice-like Principles" [Part 2] 2020-10-07 Brian Pinsky "ZFA and the Independence of Choice-like Principles" [Part 1] 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. 2020-09-30 James Holland "Iterated Forcing" [Part 4] 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] 2020-09-23 James Holland "Iterated Forcing" [Part 3] 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] 2020-09-16 James Holland "Iterated Forcing" [Part 2] 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] 2020-09-09 James Holland "Iterated Forcing" [Part 1] 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]

#### 2020 (Spring) Schedule of Speakers

 2020-04-29 Takehiko Gappo "Basic Theory of Mitchell–Steel Mice" [Part 6] Abstract: We'll finally prove the comparison lemma, which is the most important basic result in inner model theory. 2020-04-22 Brian Pinsky "Cichoń's Diagram Part 2- Random Forcing" 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 $$\aleph_\omega$$. 2020-04-15 Takehiko Gappo "Basic Theory of Mitchell–Steel Mice" [Part 5] Abstract: We'll cover the content that we didn't have time to explain the last time, i.e., a normal iteration game. 2020-04-08 Brian Pinsky "Cichoń's Diagram" [Part 1] 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. 2020-04-01 Takehiko Gappo "Basic Theory of Mitchell–Steel Mice" [Part 4] 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. 2020-03-11 James Holland "The Theory of Forcing" [Part 2] 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]. 2020-03-04 Takehiko Gappo "Basic Theory of Mitchell–Steel Mice" [Part 3] 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. 2020-02-26 James Holland "The Theory of Forcing" [Part 1] 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]. 2020-02-19 Takehiko Gappo "Basic Theory of Mitchell–Steel Mice" [Part 2] 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. 2020-02-12 James Holland "An Introduction to Measures and Ultrapowers" 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]. 2020-02-05 Takehiko Gappo "Basic Theory of Mitchell–Steel Mice" [Part 1] 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. 2020-01-29 James Holland "The Basics of Skolem Hulls, and Transitive/Mostowski Collapses" 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].