Set Theory Seminar

This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.

Slides

Video

This is a two-part talk concerning existence/non-existence of certain kinds of extensions of arbitrary models of ZF, with no regard to countability or well-foundedness of the models involved. The talk is based a recent preprint: arXiv:2406.14790v1. The results presented include the following two. In Theorem A below, N is said to be a conservative elementary extension of M if N is an elementary extension of M with the property that the intersection of every parametrically definable subset of N with M is parametrically definable in M.
Theorem A. Every model M of ZF with a definable global well-ordering has a conservative elementary extension N that contains an ordinal above all of the ordinals of M.
Theorem B. Every consistent extension of ZF has a model of power aleph_1 that has no end extension to a model of ZF.

Slides

Video

We discuss the famous open problem of determining the exact consistency strength of PFA. We present an equiconsistency between Ben Goodman's Sigma_n-Correct Proper Forcing Axiom, which implies PFA, and supercompact for C^(n-1)-cardinals under additional mild assumptions for large enough n. Without these assumptions we can prove a dichotomy resembling Woodin's HOD dichotomy with a model containing the mantle taking on the role of HOD.

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We present new results regarding the depth and Tukey spectrum of general ultrafilters and simple
$P_\lambda$-points at a measurable cardinal. In particular we prove that on a measurable cardinal there can only be a single $\lambda$ for which there exists a simple $P_\lambda$-point - this is in sharp contrast to $\omega$. Finally we will present several models in which we analyze the depth and Tukey spectrum of an ultrafilter, and their effect on generalized cardinal characteristics.

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We will consider the space of functions from $\lambda$ to $\kappa$ for various choices of $\lambda$ and $\kappa$. In the first part of the talk we define topologies on such spaces and discuss the $\mu$-meagre ideal (i.e. sets that are unions of $\mu$-many nowhere dense sets), and their associated cardinal invariants. In the second part, we will look at various ways to consider (cardinal invariants of) the null ideal on such spaces.

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We show that, given a reflecting cardinal, one can generically produce a universe of $\mathsf{BPFA}$ in which additionally the $\Sigma^1_{n+2}$-uniformization property holds for every $n$ simultaneously.

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We are interested in when are two uncountable models 'similar' - elementarily equivalent, elementarily equivalent in a stronger logic, or even isomorphic. Such equivalence is always characterized in terms of an Ehrenfeucht-Fraïssé game. We give two modifications of the standard Ehrenfeucht-Fraïssé game and discuss the logics these give rise to. The modified versions involve partitions of models and making the game longer using an Aronszajn tree as a game clock.

Video

Given a topological space $(X, \tau)$, the derived set operator $d_\tau$ maps a set $T$ to its set of limit points with respect to $\tau$. Fixing an initial topology $\tau_0$ on $X$, we can define a sequence of derived topologies $(X, \tau_0, \tau_1, \dots, \tau_\xi, \dots)$, where $\tau_\zeta \subseteq \tau_\xi$ for $\zeta \lt \xi$. This is achieved by declaring $d_{\tau_\xi}(T)$ to be open in $\tau_{\xi+1}$ and taking unions at limit stages.

In Derived Topologies on Ordinals and Stationary Reflection, Bagaria characterised the non-isolated points in the $\xi$-th derived topology on ordinals as those satisfying a strong iterated form of stationary reflection, termed $\xi$-simultaneous reflection.

Generalisations of combinatorial properties of ordinals to $\mathcal{P}_\kappa(A) := \{X \subseteq \kappa : |X| \lt \kappa\}$, where $\kappa$ is an uncountable regular cardinal and $A \subseteq \kappa$, have been widely studied. In this context, we extend the notion of higher stationarity and construct a sequence of topologies $\langle \tau_0, \tau_1, \dots \rangle$ on $\mathcal{P}_\kappa(A)$, characterising the simultaneous reflection of high-stationary subsets of $\mathcal{P}_\kappa(A)$ in terms of elements in the base of a derived topology on $\mathcal{P}_\kappa(A)$.

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