The Constructible Universe, the Naive Conception, and Intensional Logic

Speaker: 

Sean Walsh

Institution: 

Logic and Philosophy of Science, UC Irvine

Time: 

Monday, November 25, 2013 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH 440R

This talk looks at the relationship between three foundational systems: Goedel's Constructible Universe of Sets, the naive conception of set found in consistent fragments of Frege's Grundgesetze, and the intensional logic of Church's Logic of Sense and Denotation. One basic result shows how to use the constructible sets to build models of fragments of Frege's Grundgesetze from which one can recover these very constructible sets using Frege's definition of membership. This result also allows us to solve the related consistency problem and joint consistency problems for abstraction principles with limited amounts of comprehension. Another basic aim of this paper is to show how to "factor'' this result via a consistent fragment of Church's Logic of Sense and Denotation: so one may use the constructible sets to build models of Church's Logic of Sense and Denotation, from which one may then define models of the consistent fragments of Frege's Grundgesetze.
Preprint: https://www.dropbox.com/s/afhcz8bzy4pdsoc/walsh-sean-CU%2BNC%2BIL-11-19-...
 

Separating strong saturation properties of ideals on small cardinals IV

Speaker: 

Monroe Eskew

Institution: 

UCI

Time: 

Monday, October 21, 2013 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH440R

We will finish the proof that under GCH, dense ideals cannot exist at successors of singular cardinals.  Then we will outline how to separate the density property from the disjoint refinement property above aleph_1, and note remaining open questions.

Separating strong saturation properties of ideals on small cardinals III

Speaker: 

Monroe Eskew

Institution: 

UCI

Time: 

Monday, October 14, 2013 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH 440R

This is a continuation of the previous two talks, where we used large cardinals to get a normal, lambda-dense ideal on [lambda]^<kappa, where kappa is the successor of a regular cardinal, and GCH holds near kappa.  In this talk we show that the analogous statement for a successor of a singular cardinal is inconsistent.  If time permits, we will begin discussion of consistently separating certain properties at kappa>omega_1 that coincide at omega_1.

Separating strong saturation properties of ideals on small cardinals II

Speaker: 

Monre Eskew

Institution: 

UCI

Time: 

Monday, October 7, 2013 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH 440R

The topic of this talk is inspired by measure-theoretic questions raised by Ulam: What is the smallest number of countably additive, two valued measures on R such that every subset is measurable in one of them?  Under CH, the minimal answer to this question has several equivalent formulations, one of which is the maximal saturation property for ideals on aleph_1, aleph_1-density.  Our goal is to show that these equivalences are special to aleph_1.  In the second talk, we will continue with construction of normal ideals of minimal possible density on a variety of spaces from almost-huge cardinals.  This generalizes a result of Woodin.

Separating strong saturation properties of ideals on small cardinals I

Speaker: 

Monroe Eskew

Institution: 

UCI

Time: 

Monday, September 30, 2013 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH 440R

The topic of this talk is inspired by measure-theoretic questions raised by Ulam: What is the smallest number of countably additive, two valued measures on R such that every subset is measurable in one of them?  Under CH, the minimal answer to this question has several equivalent formulations, one of which is the maximal saturation property for ideals on aleph_1, aleph_1-density.  Our goal is to show that these equivalences are special to aleph_1.  In the first talk, we will show how to get normal ideals of minimal possible density on a variety of spaces from almost-huge cardinals.  This generalizes a result of Woodin.
 

Separating Square Principles at a Singular Cardinal

Speaker: 

Ryan Holben

Institution: 

UCI

Time: 

Monday, June 3, 2013 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH 440R

Starting with a cardinal which is both subcompact and measurable, we produce a model in which \square_{\kappa,2} holds but \square_\kappa fails at a singular cardinal \kappa.  We will discuss several of the essential tools used, and also several ways in which this result may be extended.
 

Pages

Subscribe to RSS - Logic Set Theory