Set Theory Seminar

In this talk, we will review some basic properties of partition cardinals under the axiom of determinacy. We will be particularly interested with the strong partition property of the first uncountable cardinal and the good coding system used to derive these partition properties. We will discuss almost everywhere behavior of functions on partition spaces of cardinals with respect to the partition measures including various almost everywhere continuity and monotonicity properties. These continuity results will be used to distinguish some cardinalities below the power set of partition cardinals. We will also use these continuity results to produce upper bounds on the ultrapower of the first uncountable cardinal by each of its partition measures, which addresses a question of Goldberg. Portions of the talk are joint work with Jackson and Trang.

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In this talk, we will review some basic properties of partition cardinals under the axiom of determinacy. We will be particularly interested with the strong partition property of the first uncountable cardinal and the good coding system used to derive these partition properties. We will discuss almost everywhere behavior of functions on partition spaces of cardinals with respect to the partition measures including various almost everywhere continuity and monotonicity properties. These continuity results will be used to distinguish some cardinalities below the power set of partition cardinals. We will also use these continuity results to produce upper bounds on the ultrapower of the first uncountable cardinal by each of its partition measures, which addresses a question of Goldberg. Portions of the talk are joint work with Jackson and Trang.

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The theory of Borel reducibility has had great success in ruling out proposed classifications in various areas of mathematics. However, this framework doesn't account for an important feature of such classifications - they are often expected to be functorial, not just respecting isomorphism but taking any homomorphism between the objects in question to a homomorphism of the invariants. I will talk about some work in progress with Filippo Calderoni, extending the framework to include functoriality and noting some differences this immediately introduces from the standard framework.

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Assuming multiple of strong cardinals, there are lots of cardinals with small degrees of strength (i.e. $\kappa$ that are $\kappa$+2-strong). We can calculate the consistency strength of these all cardinal's small degrees of strength being weakly indestructible using forcing and core model techniques in a way similar to Apter and Sargsyan's previous work. This yields some easy relations between indestructibility and Woodin cardinals, and also generalizes easily to supercompacts. I will give a proof sketches of these results.

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This is the second of two talks devoted to two properties of ultrafilters (non-principal, on omega) for which the question 'Do such ultrafilters exist?' is open. In this talk, I'll discuss the property of not being at the top of the Tukey ordering (of ultrafilters on omega). I'll start with the definition of the Tukey ordering, and I'll give an example of an ultrafilter that is 'Tukey top'. It's consistent with ZFC that some ultrafilters are not Tukey top. The examples and the combinatorial characterizations involved here are remarkably similar but not identical to examples and the characterization from the previous talk. That observation suggests some conjectures, one of which I'll disprove if there's enough time.

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This is the first of two talks devoted to two properties of ultrafilters (non-principal, on omega) for which the question 'Do such ultrafilters exist?' is open. In this talk, I'll discuss the property of being preserved by some forcing that adds new reals. Some forcings destroy all ultrafilters, and some (in fact many) ultrafilters are destroyed whenever new reals are added, but it is consistent with ZFC that some ultrafilters are preserved when some kinds of reals are added. I plan to prove some of these things and describe the rest. I'll also describe a combinatorial characterization, due to Arnie Miller, of preservable ultrafilters.

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I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the abundant atom axiom, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal $\kappa$ is supercompact if and only if for every second-order sentence true in some structure $M$ (of any size) is also true in a first-order elementary substructure $m\prec M$ of size less than $\kappa$. This is joint work with Bokai Yao. http://jdh.hamkins.org/surprising-strength-of-reflection-with-abundant-urelements-cuny-set-theory-seminar-april-2022

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Independent families are families of infinite sets of integers with the property that for any two disjoint, non-empty, finite subfamilies $A$ and $B$ of the given family, the set $\bigcap A\backslash \bigcup B$ is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, as well as their indestructibility by various forcing notions. In this talk, we will consider some recent advances in the area and point out to remaining open questions.

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Independent families are families of infinite sets of integers with the property that for any two disjoint, non-empty, finite subfamilies $A$ and $B$ of the given family, the set $\bigcap A\backslash \bigcup B$ is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, as well as their indestructibility by various forcing notions. In this talk, we will consider some recent advances in the area and point out to remaining open questions.

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We investigate which forcing notions can be embedded into a Tree-Prikry forcing. It turns out that the answer changes drastically under different large cardinal assumptions. We will focus on the class of $\kappa\text{-}$strategically closed forcings of cardinality $\kappa$, $<\kappa\text{-}$strategically closed forcings of cardinality $\kappa$ and the $\kappa\text{-}$distributive forcing notions of cardinality $\kappa$. Then we will examine distributive subforcings of the Prikry forcing of cardinality larger than $\kappa$. This is a joint work with Moti Gitik and Yair Hayut.

Slides

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In standard (ZFC) set theory, proper classes are not sets because they are too 'big' or, to put it in a formal way, because they surject onto any non-zero ordinal. We shall study this notion of 'bigness' in weaker systems of set theory, in particular those in which the Power Set Axiom fails. We will observe that in many such theories it is possible to have proper classes which are not big.

As part of this, we shall see a failed attempt to find a proper class which is not big in the theory ZF without Power Set but with Collection - which is by taking a certain symmetric submodel of a class forcing. It will turn out that this approach fails because, unlike in the set forcing case, the symmetric submodel of a class forcing need not exhibit many of the nice properties that we would expect. Notably, Collection may fail and, in fact, it is unclear which axioms need necessarily hold.

This will lead to the definition of the 'Respected Model', an alternative approach to defining a submodel of a class forcing in which Choice fails. We will investigate the properties of this new model and compare it to the symmetric version.

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In the paper 'Blowing up the power of a singular cardinal of uncountable cofinality', Gitik introduced the forcing which can violate the SCH at singular cardinals of any cofinalities, assuming that the singular cardinals are also singular in the ground model. The forcing is built up from a Mitchell increasing sequence of strong extenders, and it preserves all cardinals and cofinalities in the generic extension. In this talk, we will discuss a forcing which is built from a Mitchell increasing sequence of supercompact extenders. The forcing also violates the SCH at singular cardinals of any cofinalities which are singular in the ground model. An important feature of this forcing is that it is possible to collapse the successor of a singular cardinal, while preserving cardinals above it.

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In the paper 'Blowing up the power of a singular cardinal of uncountable cofinality', Gitik introduced the forcing which can violate the SCH at singular cardinals of any cofinalities, assuming that the singular cardinals are also singular in the ground model. The forcing is built up from a Mitchell increasing sequence of strong extenders, and it preserves all cardinals and cofinalities in the generic extension. In this talk, we will discuss a forcing which is built from a Mitchell increasing sequence of supercompact extenders. The forcing also violates the SCH at singular cardinals of any cofinalities which are singular in the ground model. An important feature of this forcing is that it is possible to collapse the successor of a singular cardinal, while preserving cardinals above it.

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We discuss different notions of a distributivity spectrum of forcings.

In the first talk, I will mainly focus on the notion of fresh functions and the corresponding spectrum. A function with domain lambda is fresh if it is new but all its initial segments are in the ground model. I will give general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing. Moreover, I will provide several examples, including well-known tree forcings on omega such as Sacks and Mathias forcing, as well as Prikry and Namba forcing to illustrate the difference between fresh functions and fresh subsets.

In the second talk, I will also discuss another ('combinatorial') distributivity spectrum; most importantly, analyzing this notion for the forcing P(omega)/fin.

This is joint work with Vera Fischer and Marlene Koelbing.

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We discuss different notions of a distributivity spectrum of forcings.

In the first talk, I will mainly focus on the notion of fresh functions and the corresponding spectrum. A function with domain lambda is fresh if it is new but all its initial segments are in the ground model. I will give general facts how to compute the fresh function spectrum, also discussing what sets are realizable as a fresh function spectrum of a forcing. Moreover, I will provide several examples, including well-known tree forcings on omega such as Sacks and Mathias forcing, as well as Prikry and Namba forcing to illustrate the difference between fresh functions and fresh subsets.

In the second talk, I will also discuss another ('combinatorial') distributivity spectrum; most importantly, analyzing this notion for the forcing P(omega)/fin.

This is joint work with Vera Fischer and Marlene Koelbing.

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