Set Theory Seminar

Let $n \lt \omega$ and $\Gamma$ either $\Pi$ or $\Sigma$. An ordinal $\alpha$ is called $\Gamma^1_n$-reflecting if for each $\beta \lt\alpha$ and each $\Gamma^1_n$-formula $\varphi$ if $L_\alpha \models \varphi(\beta)$ then there is a $\gamma \in (\beta, \alpha)$ so that $L_\gamma \models \varphi(\beta)$ where here $\models$ refers to full second order logic. The least $\Sigma^1_n$-reflecting ordinal is called $\sigma^1_n$ and the least $\Pi^1_n$-ordinal is called $\pi^1_n$. These ordinals provably exist and are countable (for all $n \lt \omega$). They arise naturally in proof theory, particularly in calibrating consistency strength of strong arithmetics and weak set theories. Moreover, surprisingly, their relation to one another relies heavily on the background set theory. If $V=L$ then for all $n \lt \omega$ we have $\sigma^1_{n+3} \lt \pi^1_{n+3}$ (due to Cutland) while under PD for all $n \lt \omega$ we have $\sigma^1_n \lt \pi^1_n$ if and only if $n$ is even (due to Kechris).
Surprisingly nothing was known about these ordinals in any model which satisfies neither $V=L$ nor PD. In this talk I will sketch some recent results which aim at rectifying this. In particular we will show that in any generic extension by any number of Cohen or Random reals, a Sacks, Miller or Laver real, or any lightface, weakly homogeneous Borel ccc forcing notion agrees with $L$ about which ordinals are $\Gamma^1_n$-reflecting (for any $n$ and $\Gamma$). Meanwhile, in the generic extension by collapsing $\omega_1$ many interesting things happen, not least amongst them that $\sigma^1_n$ and $\pi^1_n$ are increased - yet still below $\omega_1^L$ for $n > 2$. Along the way we will discuss the plethora of open problems in this area. This is joint work with Juan Aguilera.

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Let $\mathcal{E}$ denote the $\sigma$-ideal generated by closed null sets on $\mathbb{R}$. We show that the uniformity and the covering of $\mathcal{E}$ can be added to Cichoń's maximum with distinct values, more specifically, it is consistent that $\aleph_1\lt\mathrm{add}(\mathcal{N})\lt\mathrm{cov}(\mathcal{N})\lt\mathfrak{b}\lt\mathrm{non}(\mathcal{E})\lt\mathrm{non}(\mathcal{M})\lt\mathrm{cov}(\mathcal{M})\lt\mathrm{cov}(\mathcal{E})\lt\mathfrak{d}\lt\mathrm{non}(\mathcal{N})\lt\mathrm{cof}(\mathcal{N})\lt2^{\aleph_0}$ holds.

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Reinhardt embedding is an elementary embedding from $V$ to $V$ itself, whose existence was refuted under the Axiom of Choice by Kunen's famous theorem. There were attempts to get a consistent version of a Reinhardt embedding, and dropping the Axiom of Powerset is one possibility. Richard Matthews showed that $\mathsf{ZFC} + \mathrm{I}_1$ proves $\mathsf{ZFC}$ without Powerset is consistent with a Reinhardt embedding, but the embedding $j\colon V\to V$ in the Matthews' model does not satisfy the cofinality (i.e., for every set $a$ there is $b$ such that $a\in j(b)$). In this talk, I will show from $\mathsf{ZFC} + \mathrm{I}_0$ that $\mathsf{ZFC}$ without Powerset is consistent with a cofinal Reinhardt embedding.

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Borel chromatic numbers of definable graphs on Polish spaces have been studied for 25 years, starting with the seminal paper by Kechris, Solecky and Todorcevic. I will talk about some recent results about the consistent separation of uncountable Borel chromatic numbers of some particular graphs and about the Borel chromatic number of graphs related to Turing reducibility.

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For $\kappa$ a regular uncountable cardinal, we show that distributivity and base trees for $P(\kappa)/{\lt}\kappa$ of intermediate height in the cardinal interval $[\omega, \kappa)$ exist in certain models. We also show that base trees of height $\kappa$ can exist as well as base trees of various heights $\geq \kappa^+$ depending on the spectrum of cardinalities of towers in $P(\kappa)/{\lt}\kappa$. These constructions answer questions of V. Fischer, M. Koelbing, and W. Wohofsky in certain models.

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The open graph dichotomy states that the complete graph on the Cantor space is least among open graphs on analytic sets with respect to the ordering given by continuous graph homomorphisms. Ben Miller used dichotomies of this form to prove many interesting theorems in descriptive set theory. I will survey some applications to the descriptive set theory of generalised Cantor spaces. Recent results include connections with the determinacy of a class of long games.

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We will report on some recent results on the large cardinal hierarchy between the first supercompact cardinal and Vopěnka's Principle. We present various consistency results as well as a conjecture as for how the large-cardinal hierarchy of $\text{Ultimate}$-$L$ looks like at these latitudes. The main result will be the consistency with very large cardinals of a new Kimchi-Magidor configuration; namely, we will present a model where every supercompact cardinal is supercompact with inaccessible target points. This answers a question of Bagaria and Magidor. This configuration is a consequence of a new axiom (named $\mathcal A$) which regards the mutual relationship between superstrong and tall cardinals. Time permitting we shall discuss the interplay between $\mathcal A$ and $\text{Ultimate}$-$L$ and propose a few open questions.

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