NYLogic

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.

Video

(One version of) Cantor's second best theorem states that every pair of countable, dense sets of reals are isomorphic as linear orders. From the perspective of set theory it's natural to ask whether some variant of this theorem can hold consistently when 'countable' is replaced by 'uncountable'. This was shown in the affirmative by Baumgartner in 1973 who showed the consistency of 'all $\aleph_1$-dense sets of reals are order isomorphic' where a set is $\kappa$-dense for a cardinal $\kappa$ if its intersection with any open interval has size $\kappa$. The above became known as Baumgartner's axiom, denoted BA, and is an important axiom in both combinatorial set theory and set theoretic topology. BA has natural higher dimensional analogues - i.e., statements with the same relation to $\mathbb R^n$ that BA has to $\mathbb R$. It is a long standing open conjecture of Steprāns and Watson that BA implies its higher dimensional analogues.

In the talk I will describe some attempts to break the ice on this open problem mostly by looking at a family of weaker and stronger variants of BA and investigating their combinatorial, analytic and topological consequences. We will show that while some weak variants of BA have all the same consequences as BA, even weaker ones do not. Meanwhile a strengthening of BA for Baire and Polish space gives much more information.

Shelah showed that it is consistent that there are uncountable rigid non-archimedean real closed fields and, later, he and Mekler proved this in $\textbf{ZFC}$. Answering a question of Enayat, Charlie Steinhorn and I show that there are countable rigid non-archimedean real closed fields by constructing one of transcendence degree two.