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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