infinitary combinatorics without the axiom of choice

Titles and abstracts


Moti Gitik "Some speculations about Shelah's Weak Hypothesis and the PCF-conjecture"
Let λ be a singular cardinal. S. Shelah defined the following replacement for the power of λ:
pp(λ)=sup{cof(∏a/D) | a is a set of cof(λ)
regular cardinals, unbounded in λ, D is an ultrafilter on a including all co--bounded subsets of a }.
The Shelah Weak Hypothesis states that for every cardinal δ the set
{ λ < δ | λ is singular pp(λ) ≥ δ }
is at most countable.
We present new constructions of models such that:
  • there are singular κ0 < κ1 with pp0) = pp1) = κ1++,
  • there are singular κ01< ... <κn< ... (n<ω) with pp0) = pp1) = ... = ppn) = ... = (∪n<ωκn)+.
Attempts of generalizing this to ω1 many singular cardinals with relations to the PCF-conjecture will be discussed.

Steve Jackson "Some large cardinal properties under AD"
We survey the status of some large cardinal properties assuming the axiom of determinacy. We focus in particular on supercompact, Jonsson, and Rowbottom. We discuss what is known ``globally'' about these notions, and what is known locally (for the smaller cardinals). Building on some joint work with Khafizov and Lowe, we sketch a cardinal representation result which allows us to get an analysis of the (fairly small) Jonsson and Rowbottom cardinals.

Grigor Sargsyan "Core model induction beyond L(R)"
We will sketch how the core model induction works beyond L(R) and we will outline the proof of a recent theorem due to the author that CH+ω1 dense ideal on ω1 + ε (which will be explained) implies that there is a model of ADR+Θ is regular. It should be possible to combine the ideas of this talk with the ideas from Schindler's talk to prove that ADR+Θ is regular is a lower bound for all cardinals are singular.

Ralf Schindler "Mining strength from hypotheses incompatible with choice"
Over the years, the theory of core models and the core model induction has been used successfully to mine strength from set theoretical hypotheses which have their large cardinal strength deep under the surface. We shall focus on hypotheses which are incompatible with the axiom of choice, for instance ones according to which cardinal successors are singular or weakly compact. The essentials of the core model theory will be explained rather than presupposed. This is joint work with Daniel Busche.


Arthur Apter "Some remarks on the tree property in a choiceless context"
We discuss the construction of two choiceless models of ZF in which significant instances of the tree property hold. In the first of these models, DC is true, every successor cardinal is regular, every limit cardinal is singular, and every successor cardinal satisfies the tree property. In the second of these models, in which AC fails completely, there is an injective failure of SCH at ℵω, and ℵω+1 satisfies the tree property.

Ioanna Dimitriou "A modern approach to the first Gitik model"
We'll look at a set sized version of the first Gitik model, i.e., the one constructed in "All uncountable cardinals can be singular". We are going to look at a simplification of this construction. This construction is not only simpler on that it's a set sized forcing, but also on the actual gears of the forcing itself. It uses tree-Prikry forcings and variations of them that interact with each other as little as possible. We'll see as many properties of it as times allows and in the end we'll see some conjectures that will hopefully carry us into the discussion session.

Jip Veldman "On mutual stationarity properties below ℵω without AC"
In this talk I will present some equiconsistency results for mutual stationarity below ℵω without the axiom of choice. This is joint work with Arthur Apter and Peter Koepke.

Philip Welch "On free subsets of ℵω and a question of Pereira"
We consider the following question of Pereira: given F:[ℵω] → ℵω is there an internally approachable N, which is an elementary substructure of Hω+1 of size some ℵn, which contains F, but whose characteristic function χN has range containing a free subset for F?
The question relates to pcf theory, and may be simply be asking for something inconsistent. We observe that if it were to hold then, then the Mitchell core model for measures contains measurables of small Mitchell order.
To state a Conjecture:
A positive answer is consistent with ZF starting from a model with a single measurable.