Mathematical, foundational and computational aspects of the higher infinite

Programme theme

The goals of set theory are the analysis of the structure of the Higher Infinite, i.e. Cantor's set-theoretic universe and the elucidation of the nature of infinite mathematical objects and their role in foundational issues underlying mathematics. Moreover, the current standard system of set theory, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC), is the usual framework for a large part of mathematics.

Current set-theoretic research on infinity focuses on the following three broad areas: large Cardinals and inner model theory, descriptive set-theoretic methods and classification problems, and infinite combinatorics.

The programme HIF will connect these three main strands of set-theoretic research and other fields of set theory to the wider scope of mathematics, to research in the foundations of mathematics, including some philosophical issues, and to research on computational issues of infinity, e.g. in theoretical computer science and constructive mathematics.

The following topics are a non-exclusive list of important examples of relevant fields for the research done in the programme HIF:

The structure of definable subsets of the continuum
Infinite combinatorics, forcing, and large cardinals
Inner models of large cardinals and aspects of determinacy
Applications of set theory to other areas of mathematics
Constructive set theory and new models of computation
Set theory and the foundations of mathematics

Three workshops are planned during the programme: The first one (24-28 August 2015) will be the 5th European Set Theory Conference. The second workshop, entitled "New challenges in iterated forcing" will be a Satellite Meeting held at the University of East Anglia in Norwich (2-6 November 2015). A final workshop will take place on 14-18 December 2015.

Another UK event closely related to the HIF topics are the LMS Invited Lectures 2014 given by Jouko Väänänen at UEA, April 14-17, 2014.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
Derived topologies on ordinals and stationary reflection Authors: Joan Bagaria Download PDF	2015	HIF
Mathias and set theory Authors: A Kanamori Download PDF	2015	HIF
Homogeneous families on trees and subsymmetric basic sequences Authors: C Brech, J Lopez-Abad, S Todorcevic Download PDF	2015	HIF
Dependent choice, properness, and generic absoluteness Authors: Asaf Karagila, David Aspero Download PDF	2015	HIF
THE UNIVERSAL HOMOGENEOUS TRIANGLE-FREE GRAPH HAS FINITE BIG RAMSEY DEGREES Authors: N. Dobrinen Download PDF	2015	HIF
Towards computable analysis on the generalised real line Authors: Hugo Nobrega, Lorenzo Galeotti Download PDF	2015	HIF
Cofinalities of Marczewski-like ideals Authors: Yurii Khomskii, Wolfgang Wohofsky, Jörg Brendle Download PDF	2015	HIF
The rearrangement number Authors: Dr Will Brian, et al., Professor Andreas Blass, Professor Jörg Brendle, Joel David Hamkins Download PDF	2015	HIF
Filter-Laver Measurability Authors: Dr Yuri Khomskii (Universität Hamburg) Download PDF	2015	HIF
Cofinalities of Marczewski-like ideals Authors: Prof. Jörg Bendle (Graduate School of System Informatics Kobe University), Dr Yurii Khomskii (Hamburg University), Dr Wolfgang Wohofskyy (Hamburg University) Download PDF	2015	HIF
Power Kripke-Platek set theory and the axiom of choice Authors: Michael Rathjen Download PDF	2015	HIF
Generic Vopěnka's principle, remarkable cardinals, and the weak proper forcing axiom Authors: Joan Bagaria, Dr Victoria Gitman (The City University of New York), Prof. Ralf Schindler (Institut für mathematische Logik und Grundlagenforschung) Download PDF	2015	HIF
The modal logic of inner models Authors: T Inamdar, B Lowe Download PDF	2015	HIF
Partitions and conservativity Authors: A Blass Download PDF	2015	HIF
Almost disjoin refinements and mixing reals Authors: B Farkas, Y Khomskii, Z Vidnyanszky Download PDF	2015	HIF
Creature forcing and topological Ramsey spaces Authors: N Dobrinen Download PDF	2015	HIF
Infinite dimensional Ellentuck spaces and Ramsey-classification theorems Authors: N Dobrinen Download PDF	2015	HIF
G$\delta$$\sigma$-games and generalized computation Authors: PD Welch Download PDF	2015	HIF
Obtaining Woodin's cardinals Authors: PD Welch Download PDF	2015	HIF
On hidden variables: value and expectation no-go theorems Authors: A Blass, Y Gurevich Download PDF	2015	HIF
Open determinacy for class games Authors: V Gitman, JD Hamkins Download PDF	2015	HIF
A position in infinite chess with game value $\omega$$^4$ Authors: CDA Evans, JD Hamkins, NL Perlmutter Download PDF	2015	HIF
Upward closure and amalgamation in the generic multiverse of a countable model of set theory Authors: JD Hamkins Download PDF	2015	HIF
Game characterizations and lower cones in the Weihrauch degrees Authors: H Nobrega, A Pauly Download PDF	2015	HIF
Measures on Suslinean spaces Authors: P Borodulin-nadzieja, G Plebanek Download PDF	2015	HIF
A multiplication operation for the hierachy of norms Authors: AC Block, B Lowe Download PDF	2015	HIF
Normal measures and strongly compact cardinals Authors: AW Apter Download PDF	2015	HIF
Forcing the truth of a weak form of Schanuel's conjecture Authors: M Viale Download PDF	2015	HIF
$\it C$-algebras and B-names for complex numbers Authors:* A Vaccaro, M Viale Download PDF	2015	HIF
The minimal size of infinite maximal antichains in direct products of partial orders Authors: MS Kurilic Download PDF	2015	HIF
Recognizable sets and Woodin cardinals: computation beyond the constructible universe Authors: M Carl, P Schlicht, P Welch Download PDF	2015	HIF
Inner models from extended logics Authors: J Kennedy, M Magidor, J Vaananen Download PDF	2015	HIF
On properties of compacta that do not reflect in small continuous images Authors: M Magidor, G Plebanek Download PDF	2015	HIF
Seperating the fan theorem and its weakenings II Authors: RS Lubarsky Download PDF	2015	HIF
Philosophy or not? The study of cultures and practices of mathematics Authors: B Lowe Download PDF	2015	HIF
A candidate for the generalised real line Authors: L Galeotti Download PDF	2015	HIF
Restrictiveness relative to notions of interpretation Authors: L Incurvati, B Lowe Download PDF	2015	HIF
The quandry of quandles: The Borel completeness of a knot invariant Authors: AD Brooke-taylor, SK Miller Download PDF	2015	HIF
Ascending paths and forcings that specialize higher Aronszajn trees Authors: P Lucke Download PDF	2015	HIF
The $\kappa$$^+$-antichain property for ($\kappa$,1)-simplified morasses Authors: C Morgan Download PDF	2015	HIF
$\Sigma$$_1$($\kappa$)-Definable subsets of H($\kappa$$^+$) Authors: P Lucke, R Schindler, P Schlicht Download PDF	2015	HIF
Diagnonalising an ultrafilter and preserving a $\rho$-point Authors: H Mildenberger Download PDF	2015	HIF
The higher sharp I Authors: Y Zhu Download PDF	2015	HIF
Exact completetion of path categories and algebraic set theory Authors: B Van Den Berg, I Meordijk Download PDF	2015	HIF
Measures and fibers Authors: P Borodulin-nadzieja Download PDF	2015	HIF
Measures and slaloms Authors: P Borodulin-nadzieja, T Inamdar Download PDF	2015	HIF
Singularizing successor cardinals by forcing Authors: D Adolf, AW Apter, P Koepke Download PDF	2015	HIF
The Haplern-Läuchli theorem at a measurable cardinal Authors: N Dobrinen, D Hathaway Download PDF	2015	HIF

Workshops

HIFW01

Seminars

Monday 24th August 2015
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from John Toland (INI Director)	No Room Required
10:00 to 11:00	101 Years of Modern Set Theory: Felix Hausdorff's "Foundations of Set Theory"	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	The tree property (session 1)	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:00	What is a forcing extension (of V)?	Room 2
13:30 to 14:00	A generalisation of closed unbounded sets	Room 1
14:00 to 14:30	Hyperclass Forcing in Morse Kelley Set Theory	Room 2
14:00 to 14:30	Topological representation of lattice homomorphisms	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Around the definability of mad families	Room 1
16:00 to 17:00	The Hurewicz dichotomy for generalized Baire spaces	Room 1
17:00 to 18:00	Wine Reception	No Room Required

Tuesday 25th August 2015
09:00 to 10:00	The Search for Deep Inconsistency	Room 1
10:00 to 11:00	Local Ramsey Theory in abstract spaces	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	The tree property (session 2)	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:00	Generic I0 at $\aleph_\omega$	Room 1
14:00 to 14:30	Forcing, regularity properties and the axiom of choice	Room 1
14:00 to 14:30	Forcing failures of covering in HOD	Room 2
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Symmetries	Room 1
16:00 to 17:00	Failures of the Forcing Theorem	Room 1

Wednesday 26th August 2015
09:00 to 10:00	Hausdorff Medal Award	Room 1
10:00 to 11:00	Hausdorff Medal Lecture	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	The tree property (session 3)	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:00	Universally Baire subsets of $2^{\kappa}$	Room 1
13:30 to 14:00	Cofinalities of Marczewski Ideals	Room 2
14:00 to 14:30	Partition Relation Equiconsistent with $\exists \kappa(o(\kappa) = \kappa^+)$	Room 2
14:00 to 14:30	On the class of perfectly null sets and its transitive version	Room 1

Thursday 27th August 2015
09:00 to 10:00	Mathias and Set Theory	Room 1
10:00 to 11:00	The relation of attack	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Happy Families and Their Relatives	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:00	Selective properties of ideals	Room 1
13:30 to 14:00	Connecting topological dimension theory and recursion theory	Room 2
14:00 to 14:30	Chain conditions, layered partial orders and weak compactness	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Prikry type sequences: a composition of interconnected results	Room 1
16:00 to 17:00	A super-Dowker filter	Room 1
19:00 to 22:00	Conference Dinner at Corpus Christi College	No Room Required

Friday 28th August 2015
09:00 to 10:00	Infinite dimensional Ellentuck spaces	Room 1
10:00 to 11:00	Ultrafilters without p-point quotients	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Delta_1-definability of the non-stationary ideal	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:00	On ideal equal convergence	Room 1
14:00 to 14:30	The size of conjugacy classes of automorphism groups	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	The higher sharp	Room 1
16:00 to 17:00	Indestructible remarkable cardinals	Room 1

Tuesday 1st September 2015
13:00 to 14:00	A look into generalised stability	Room 1
14:00 to 15:00	In Pursuit of the Missing Premise in First Order Logic	Room 1
15:00 to 15:30	Coffee Break	No Room Required
15:30 to 16:30	Models of Intuitionistic Zermelo-Frankel Set Theory based on Scott's $D_{\infty}$	Room 1
16:30 to 17:30	Generic Automorphisms of a Hrushovski construction	Room 1

Wednesday 2nd September 2015
10:00 to 11:00	An application of Model Theory to Real Algebraic Geometry	Room 1
11:00 to 11:30	Coffee Break	No Room Required
11:30 to 12:30	Applying constructive analysis and topology in game theory	Room 1
13:30 to 14:40	Weak Partition Relations and Conservative Elementary Extensions	Room 1
14:40 to 15:10	Self-determined sets of sentences	Room 2
14:40 to 15:10	AEC tameness from large cardinals via category theory	Room 1
15:10 to 15:40	Where are the women in medieval logic?	Room 2
15:10 to 15:40	A variant of $\Pi^0_1$ class	Room 1
15:40 to 16:00	Coffee Break	No Room Required
16:00 to 17:00	Many-dimensional Modal Logics	Room 1
17:00 to 18:00	Drinks Reception	No Room Required

Thursday 3rd September 2015
09:00 to 10:00	Descriptive Graph Combinatorics	Room 1
09:00 to 17:00	HIFW04	Room 2
09:00 to 19:00	HIFW04 - Please note the group will need room till 1900 hrs	Room 1
10:00 to 11:00	A Multilinear Algebraic Semantics for Natural Language	Room 1
11:00 to 11:20	Coffee Break	No Room Required
11:20 to 12:20	Descriptive set theory, endofunctors & hypercomputation	Room 1
12:20 to 13:30	Lunch	No Room Required
13:30 to 14:30	Extending the Syllogistic	Room 1
14:40 to 15:10	On a Generalization of the Revision-theoretic Approach to Strategic Rationality	Room 1
15:10 to 15:40	First-order Proofs Without Syntax	Room 1
15:10 to 15:40	Partition Relation for linear Orders without the Axiom of Choice	Room 2
15:40 to 16:00	Coffee Break	No Room Required
16:00 to 17:00	Ehrenfeucht Principles in Set Theory	Room 1
17:15 to 18:15	BLC General Meeting	Room 1
19:00 to 19:30	Dinner at Robinson College	No Room Required

Friday 4th September 2015
09:00 to 15:00	HIFW04	Room 1
09:00 to 10:00	Local Names	Room 1
09:00 to 17:00	No longer required - MS unable to delete 18/08/15	Discussion Room
10:10 to 10:40	Integration on the Surreals	Room 1
10:10 to 10:40	Feedback Computability	Room 2
10:40 to 11:10	QBF proof complexity	Room 1
10:40 to 11:10	Computably extendible order types	Room 2
11:10 to 11:30	Coffee Break	No Room Required
11:30 to 12:30	Forcing with large continuum	Room 1
12:30 to 13:30	Lunch	No Room Required
13:30 to 14:30	Constructing Quasiminimal Functions	Room 1

Tuesday 8th September 2015
15:00 to 16:00	Another proof of the failure of a higher forcing axiom	Room 2

Thursday 10th September 2015
15:00 to 16:00	On the width of wqos	Room 2

Tuesday 15th September 2015
15:00 to 16:00	$\Sigma^0_3$ determinacy and friends	Room 2

Thursday 17th September 2015
15:00 to 16:00	Linking set theory to economics	Room 2

Tuesday 29th September 2015
15:00 to 16:00	Modal logics of the generic multiverse	Room 2
16:00 to 17:00	Aspects of generalizing the concept of strong measure zero	Room 2

Thursday 1st October 2015
15:00 to 16:00	The Silence	Room 2

Monday 5th October 2015
16:00 to 17:00	Beyond the infinite: Rothschild Distinguished Visiting Professor Lecture	Room 1

Tuesday 6th October 2015
15:00 to 16:00	Reversibility of Definable Relations	Room 2

Friday 9th October 2015
12:30 to 13:25	Computable analysis and games in descriptive set theory	Room 2
13:30 to 14:25	Min-Max theorems in infinite combinatorics	Room 2
14:40 to 15:35	Saturated Boolean Ultrapowers	Room 2
15:50 to 16:45	Uncountably many maximal-closed subgroups of Sym(N) via reducts of Henson digraphs	Room 2

Saturday 10th October 2015
11:00 to 11:55	Weihrauch degrees for generalized Baire space	Room 1
13:00 to 13:55	Topological Ramsey theory of countable ordinals	Room 1
14:00 to 14:55	Free groups and the Axiom of Choice	Room 1

Thursday 22nd October 2015
15:30 to 16:30	Universal graphs and their variations	Room 2

Tuesday 27th October 2015
15:00 to 16:00	Dual Ramsey theory on trees	Room 2
16:00 to 17:00	Order types of chains of Borel sets and Baire functions	Room 2

Wednesday 28th October 2015
11:00 to 12:00	Ramsey Determinacy of adversarial Gowers games	Room 2

Thursday 29th October 2015
15:00 to 16:00	Approximate Ramsey properties of matrices and finite dimensional normed spaces	Room 2
16:00 to 17:00	Consecutively large families below the first inaccessible cardinal	Room 2

Tuesday 10th November 2015
15:00 to 16:00	On two problems on Boolean algebras and finitely additive measures	Room 2

Tuesday 17th November 2015
15:00 to 16:00	Reflection numbers of some combinatorial and topological properties	Room 2

Thursday 19th November 2015
15:00 to 16:00	The surreal numbers	Room 1

Tuesday 1st December 2015
15:00 to 16:00	Forcing the truth of a weak form of Schanuel's conjecture	Room 2

Tuesday 8th December 2015
15:00 to 16:00	The nature of measurement in set theory	Room 2

Thursday 10th December 2015
15:00 to 16:00	Some more on $G_\delta\sigma$ determinacy and generalized recursion	Room 2

Monday 14th December 2015
09:30 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from Christie Marr (INI Deputy Director)	No Room Required
10:00 to 11:00	Ramsey theory in topological dynamics	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
12:00 to 13:30	Lunch Break	No Room Required
13:30 to 14:30	Approximate Ramsey properties of Matrices	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 15:30	The unreasonable effectiveness of Nonstandard Analysis	Room 1
15:30 to 16:00	Break	No Room Required
16:00 to 17:00	A revision theory for type-free probability	Room 1
17:00 to 18:00	Welcome Wine Reception	No Room Required

Tuesday 15th December 2015
10:00 to 11:00	On relating strong type theories and set theories	Room 1
10:00 to 11:00	On relating strong type theories and set theories	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Inaccessible cardinals and accessible categories	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:30	The Pinning Down Number and Cardinal Arithmetic	Room 1
14:30 to 14:45	Afternoon Tea	No Room Required
15:45 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	TBA	Room 1
18:00 to 19:00	Cambridge University Press Wine Reception CUP Book Shop, 1 Trinity Street	No Room Required

Wednesday 16th December 2015
10:00 to 11:00	Well-quasi-orderings for progam analysis and computational complextiy	Room 1
10:00 to 11:00	Well-quasi-orderings for progam analysis and computational complextiy	Room 1
11:00 to 11:30	Morning coffee	No Room Required
11:30 to 12:00	Infinite Matroids and Pushdown Automata on Infinite Words	Room 1
12:00 to 12:30	Weihrauch degrees of determinacy	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:30	Computing beyond Constructibility: The Recognizability Strength of Ordinal Time Machines	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Long and short recursive constructions---cardinal invariants and parametrized diamonds	Room 1

Thursday 17th December 2015
10:00 to 11:00	Set Theory and Automata Theory	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Context-Dependent Deterministic Parallel Feedback Turing Computability	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:30	An Introduction to infinite matroids	Room 1
14:30 to 14:45	Afternoon Tea	No Room Required
14:45 to 15:45	Determinacy in Infinite Matroids	Room 1
15:45 to 16:00	Afternoon Tea	No Room Required
16:00 to 17:00	Pairwise Sums in the Reals	Room 1
19:00 to 22:00	Conference Dinner at Corpus Christi College	No Room Required

Friday 18th December 2015
10:00 to 11:00	Another proof of the Jayne-Rogers theorem	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Borel Matchings and equidecompositions	Room 1
12:30 to 13:30	Lunch Break	No Room Required
13:30 to 14:30	Set theory and algebraic topology	Room 1
13:30 to 14:30	Set theory and algebraic topology	Room 1
14:30 to 17:00	INI End-of-the-Year Party	No Room Required

Mathematical, foundational and computational aspects of the higher infinite

Organisers

Joan Bagaria ICREA and Universitat Autonoma de Barcelona
Mirna Dzamonja University of East Anglia
Benedikt Löwe Universiteit van Amsterdam; Universität Hamburg

Mathematical, foundational and computational aspects of the higher infinite

Programme theme

Preprints

Derived topologies on ordinals and stationary reflection

Mathias and set theory

Homogeneous families on trees and subsymmetric basic sequences

Dependent choice, properness, and generic absoluteness

THE UNIVERSAL HOMOGENEOUS TRIANGLE-FREE GRAPH HAS FINITE BIG RAMSEY DEGREES

Towards computable analysis on the generalised real line

Cofinalities of Marczewski-like ideals

The rearrangement number

Filter-Laver Measurability

Cofinalities of Marczewski-like ideals

Power Kripke-Platek set theory and the axiom of choice

Generic Vopěnka's principle, remarkable cardinals, and the weak proper forcing axiom

The modal logic of inner models

Partitions and conservativity

Almost disjoin refinements and mixing reals

Creature forcing and topological Ramsey spaces

Infinite dimensional Ellentuck spaces and Ramsey-classification theorems

G$\delta$$\sigma$-games and generalized computation

Obtaining Woodin's cardinals

On hidden variables: value and expectation no-go theorems

Open determinacy for class games

A position in infinite chess with game value $\omega$$^4$

Upward closure and amalgamation in the generic multiverse of a countable model of set theory

Game characterizations and lower cones in the Weihrauch degrees

Measures on Suslinean spaces

A multiplication operation for the hierachy of norms

Normal measures and strongly compact cardinals

Forcing the truth of a weak form of Schanuel's conjecture

$\it C$*-algebras and B-names for complex numbers

The minimal size of infinite maximal antichains in direct products of partial orders

Recognizable sets and Woodin cardinals: computation beyond the constructible universe

Inner models from extended logics

On properties of compacta that do not reflect in small continuous images

Seperating the fan theorem and its weakenings II

Philosophy or not? The study of cultures and practices of mathematics

A candidate for the generalised real line

Restrictiveness relative to notions of interpretation

The quandry of quandles: The Borel completeness of a knot invariant

Ascending paths and forcings that specialize higher Aronszajn trees

The $\kappa$$^+$-antichain property for ($\kappa$,1)-simplified morasses

$\Sigma$$_1$($\kappa$)-Definable subsets of H($\kappa$$^+$)

Diagnonalising an ultrafilter and preserving a $\rho$-point

The higher sharp I

Exact completetion of path categories and algebraic set theory

Measures and fibers

Measures and slaloms

Singularizing successor cardinals by forcing

The Haplern-Läuchli theorem at a measurable cardinal

Workshops

The 5th European set theory conference

New challenges in iterated forcing

The role of the higher infinite in mathematics and other disciplines

Seminars

Organisers

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