Online activities 17-23 August 2020

Bar-Ilan-Jerusalem Set Theory Seminar
Time: Wednesday, August 19, 11:00am Israel Time (10:00 CEST)
Speaker: Uri Abraham, Ben-Gurion University
Title: Coding well ordering of the reals with ladders, part 5 
Abstract: Results from the 2002 paper “Coding with Ladders a Well Ordering of the Reals” by Abraham and Shelah.
Information: contact Menachem Magidor, Asaf Rinot or Omer Ben-Neria ahead of time for the zoom link.

CUNY Logic Seminar (MOPA)
Time: Wednesday, August 19, 14:00 New York time (20:00 CEST) – note the time
Speaker: Leszek Kołodziejczyk, University of Warsaw
Title: Ramsey’s Theorem over RCA_0*
Abstract: The usual base theory used in reverse mathematics, RCA0, is the fragment of second-order arithmetic axiomatized by Δ01 comprehension and Σ01 induction. The weaker base theory RCA∗0 is obtained by replacing Σ01 induction with Δ01 induction (and adding the well-known axiom exp in order to ensure totality of the exponential function). In first-order terms, RCA0 is conservative over IΣ1 and RCA∗0 is conservative over BΣ1+exp. 
Some of the most interesting open problems in reverse mathematics concern the first-order strength of statements from Ramsey Theory, in particular Ramsey’s Theorem for pairs and two colours. In this talk, I will discuss joint work with Kasia Kowalik, Tin Lok Wong, and Keita Yokoyama concerning the strength of Ramsey’s Theorem over RCA∗0.Given standard natural numbers n,k≥2, let RTnk stand for Ramsey’s Theorem for k-colourings of n-tuples. We first show that assuming the failure of Σ01 induction, RTnk is equivalent to its own relativization to an arbitrary Σ01-definable cut. Using this, we give a complete axiomatization of the first-order consequences of RCA∗0+RTnk for n≥3 (this turns out to be a rather peculiar fragment of PA) and obtain some nontrivial information about the first-order consequences of RT2k. Time permitting, we will also discuss the question whether our results have any relevance for the well-known open problem of characterizing the first-order consequences of RT22 over the traditional base theory RCA0.
Information: The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

CUNY Set Theory Seminar
Time: Friday, August 21, 14:00 New York time (20:00 CEST)
Speaker: Dan Hathaway University of Vermont
Title: A relative of ZF+DC+‘ω1 is measurable’
Abstract: Let Φ be the statement that for any function f:ω1×ω1→ω, there are functions g1,g2:ω1→ω such that for all (x,y)∈ω1×ω1, we have f(x,y)≤max {g1(x),g2(y)}. We will show that Φ follows from ZF+DC+‘ω1 is measurable’. On the other hand using core models, we will show that Φ+‘the club filter on ω1 is normal’ implies there are inner models with many measurable cardinals. We conjecture that Φ and ZF+DC+‘ω1 is measurable’ have the same consistency strength. The research is joint with Francois Dorais at the University of Vermont.
Information: Please email Victoria Gitman (vgitman@nylogic.org) for meeting id (this talk will have a different meeting ID!).

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