Kobe Set Theory Seminar
Time: Monday, 5 June, 16:30 local time (09:30 CEST)
Speaker: Sakaé Fuchino
Title: Resurrection and Maximality under the tightly Laver-generically ultrahuge cardinal (2/2)
Abstract: A (definable) class P of posets is said to be iterable if ① P is closed with respect to forcing equivalence (i.e. if ℙ∈P and ℙ∼ℙ′ then ℙ′∈P ), ② closed wrt restriction (i.e. if ℙ∈P then ℙ↾𝕡∈P for any 𝕡∈ℙ ), and, ③ for any ℙ∈P and ℙ-name ℚ˙, ⫦ℙ“ℚ˙∈P” implies ℙ∗ℚ˙∈P.
For an iterable class P of posets, a cardinal κ is said to be P-Laver-generically supercompact if, for any λ≥κ and ℙ∈P, there is a ℙ-name ℚ˙ with ⫦ℙ“ℚ˙∈P” such that, for (𝖵,ℙ∗ℚ˙)-generic ℍ, there are j, M⊆𝖵[ℍ] with
(a) j:𝖵≺_κ M, (b) j(κ)>λ, and (c) ℙ, ℙ∗ℚ˙, ℍ, j′′λ∈M.
κ is said to be tightly P-Laver-generically supercompact if additionally (d) ∣∣ℙ∗ℚ˙∣∣≤j(κ) holds.
Similarly, we can also define (tightly) P-Laver-generic versions of super almost-huge, superhuge, and ultrahuge cardinals.
In [ II ] it is shown that the existence of P-Laver-gen. supercompact cardinal (tightly P-Laver gen. superhuge in the case P = ccc posets) for a reasonable P highlights the situations with the continuum being ① ℵ1, ② ℵ2 or ③ very large.
In particular with P being the class of all ① σ-closed posets, ② semi-proper posets, or ③ ccc-posets, the existence of P-Laver-gen. supercompact cardinal (or tightly P-Laver gen. superhuge in the case P = ccc posets) implies a double-plused version of forcing axiom for the respective P and strong reflection properties down to less than κ_refl:=max{ℵ2,2ℵ0} compatible with the forcing axiom.
In this talk we shall prove that the existence of tightly P-Laver-generically superhuge cardinal implies the boldface version of Resurrection Axiom ([Hamkins-Johnstone 1], [Hamkins-Johnstone 2] ) for P over H(κ_refl).
We further show that the existence of tightly P-Laver-generically ultrahuge cardinal implies the Unbounded Resurrection Axiom of Tsaprounis ([Tsaprounis]) for P and strong version of local maximality principle ((slightly?) stronger than the one mentioned in [Minden]).
References.
[ I ] S.F., A. Ottenbreit Maschio Rodrigues, and H. Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, Archive for Mathematical Logic, Vol.60, 1-2, (2021), 17–47. https://fuchino.ddo.jp/papers/SDLS-x.pdf
[ II ] —–, Strong downward Löwenheim-Skolem theorems for stationary logics, II — reflection down to the continuum, Archive for Mathematical Logic, Vol.60, 3-4, (2021), 495–523. https://fuchino.ddo.jp/papers/SDLS-II-x.pdf
[Hamkins-Johnstone 1] Joel David Hamkins, and Thomas A.Johnstone, Resurrection axioms and uplifting cardinals, Archive for Mathematical Logic, Vol.53, Iss.3-4, (2014), 463-485.
[Hamkins-Johnstone 2] —–, Strongly uplifting cardinals and the boldface resurrection axioms, Archive for Mathematical Logic volume 56, (2017), 1115-1133.
[Minden] Kaethe Minden, Combining resurrection and maximality, The Journal of Symbolic Logic, Vol. 86, No. 1, (2021), 397–414.
[Tsaprounis 1] Konstantinos Tsaprounis, On resurrection axioms, The Journal of Symbolic Logic, Vol.80, No.2, (2015), 587–608.
[Tsaprounis 2] —–, Ultrahuge cardinals, Mathematical Logic Quarterly, Vol.62, No.1-2, (2016), 1–2.
Information: Please see the seminar webpage. This talk will be given in hybrid format. Please email Hiroshi Sakai in advance for the zoom information.
Baltic Set Theory Seminar
Time: Tuesday, 6 June, 15:00-16:30 CEST
Speaker: Several
Title: Baltic Set Theory Seminar
Abstract: This is a learning seminar, the goal is to actually go over proofs and more or less understand them. Discussions are encouraged. The topic of the seminar is the following:
1. Matteo Viale, Generic absoluteness theorem for the omega_1 Chang model conditioned to MM^{+++}.
2. Paul Larson, A course on AD^+
Information: Please see the seminar webpage.
Leeds Models and Sets Seminar
Time: Wednesday, 7 June, 13:45-15:00 local time (14:45-16:00 CEST)
Speaker: Richard Matthews, Université Paris-Est Créteil Val de Marne
Title: Very large set axioms over Constructive Set Theories
Abstract: One of the main areas of research in set theory is the study of large cardinal axioms and many of these can be characterised by the existence of elementary embeddings with certain properties. The guiding principle is then that the closer the domain and co-domain of the embedding is to the universe, the stronger the resulting large cardinal axiom. This leads naturally to the question of whether there is an elementary embedding of the universe into itself which is not the identity, and the least ordinal moved by such an embedding is known as a Reinhardt cardinal. While Kunen famously proved that no such embedding can exist if the universe satisfies ZFC, it is an open question in many subtheories of ZFC, most notably ZF (without Choice).
In this talk we will study elementary embeddings in the weaker context of intuitionistic set theories, that is set theories without the law of excluded middle. We shall observe that the ordinals can be very ill-behaved in this setting and therefore we will reformulate large cardinals by instead looking for large sets which capture the desired structural properties. We shall investigate the consistency strength of analogues to measurable cardinals, Reinhardt cardinals and many other similar ideas in terms of the standard ZFC large cardinal hierarchy.
This is joint work with Hanul Jeon.
Information: Please see the seminar webpage.
Caltech Logic Seminar
Time: Wednesday, 7 June, 12:00am-13:00pm Pacific time (21:00-22:00 CEST)
Speaker: Clinton Conley, CMU
Title: Borel asymptotic dimension and hyperfiniteness
Abstract: We introduce a “purely Borel” version of Gromov’s notion of asymptotic dimension, and show how to use it to establish hyperfiniteness of various equivalence relations. Time permitting, we discuss hyperfiniteness of orbit equivalence relations of free actions of lamplighter groups. This is joint work with Jackson, Marks, Seward, and Tucker-Drob.
Information: Please see the seminar webpage.
Cross-Alps Logic Seminar
Time: Friday, 9 June, 16.00-17.00 CEST
Speaker: Ulrich Kohlenbach, Technische Universität Darmstadt
Title: Proof mining: Recent developments
Abstract: In this talk we survey some recent developments in the project of applying proof-theoretic transformations to obtain new quantitative and qualitative information from given proofs in areas of core mathematics such as nonsmooth optimization, geodesic geometry and ergodic theory. We will discuss some of the following items:
(1) Proof mining in the context of set-valued monotone and accretive operators with applications in nonsmooth optimization such as inconsistent feasibility theorems (partly joint work with Nicholas Pischke).
(2) Recent linear rates of asymptotic regularity as well as rates of metastability for Tikhonov-regularization methods (joint work with Horaţiu Cheval and Laurenţiu Leuştean).
(3) The extraction of uniform rates of convergence for the ε-capture in the Lion-Man game in a general geodesic setting from a proof that made iterated use of sequential compactness arguments (i.e. arithmetical comprehension). The extraction also qualitatively generalizes previously known results (joint work with Genaro López-Acedo and Adriana Nicolae).
(4) Recent applications to ergodic theory (joint work with Anton Freund).
Information: The event will stream on the Webex platform. Please write to luca.mottoros [at] unito.it for the link to the event.
European Set Theory Society 