# Welcome to New York Logic

Welcome to New York Logic!

Aug 23 Tuesday Aug 24 Wednesday Aug 25 Thursday Aug 26 Friday Aug 27 Saturday Aug 28 Sunday Aug 29 Monday

go to the Calendar

## Upcoming talks and events:

Set theory seminarFriday, September 2, 201610:00 amGC 6417

# Set-theoretic geology and the downward-directed grounds hypothesis

The City University of New York

Forcing is often viewed as a method of constructing larger models extending a given model of set theory. The topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model. Thus, an inner model $W\subset V$ is a ground of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb Q\in W$. Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

CUNY Logic WorkshopFriday, September 2, 20162:00 pmGC 6417

# Title TBA

The City University of New York
Set theory seminarFriday, September 9, 201610:00 amGC 6417

# Set-theoretic geology and the downward-directed grounds hypothesis: part II

The City University of New York

I will continue presenting Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. See the main abstract at Set-theoretic geology and the downward directed ground hypothesis.

CUNY Logic WorkshopFriday, September 9, 20162:00 pmGC 6417

# Title TBA

Rutgers University
Set theory seminarFriday, September 23, 201610:00 amGC 6417

# On the non-existence and definability of mad families

Hebrew University of Jerusalem

By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+”there are no mad families” is actually equiconsistent with ZFC. I’ll present the ideas behind the proof in the first part of the talk.

In the second part of the talk, I’ll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I’ll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I’ll show how large cardinals must be involved in such a solution.

This is joint work with Saharon Shelah.

CUNY Logic WorkshopFriday, September 23, 20162:00 pmGC 6417

# TBA

National University of Singapore
Friday, September 30, 2016

# Cherlin Weekend at Rutgers

Rutgers University, Hill 705.

Rutgers University will hold a conference in honor of Professor Gregory Cherlin on September 30 – October 2, 2016. Details are available here.

CUNY Logic WorkshopFriday, October 7, 20162:00 pmGC 6417

# Title TBA

York University
NERDS: New England Recursion & Definability SeminarSunday, November 6, 2016

# Autumn 2016 NERDS at Wellesley College

Wellesley College

The Autumn 2016 meeting of NERDS, the New England Recursion and Definability Seminar, will take place on Sunday, November 6 at Wellesley College, in Wellesley, MA. Details will be posted here as they become available.

CUNY Logic WorkshopFriday, November 11, 20162:00 pmGC 6417

# Title TBA

University of Auckland