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Seminar: 2015-2016

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Unless otherwise indicated, all meetings take place on Tuesdays at 4:30PM at 380-381T

The seminar is led by Professor Solomon Feferman

Autumn

This quarter and next we shall largely be making use of materials from the Exploring the Frontiers of Incompleteness (EFI) workshops held at Harvard in 2011-2012. The EFI site provides background materials, a series of eleven individual lectures and/or individual workshop materials. Some of the topics discussed are the incompleteness phenomena for set theory, the programs for new axioms--in particular large cardinal axioms--to overcome incompleteness, the axiom of determinacy, and the continuum problem. This material is of great active logical and philosophical interest. It will be assumed that participants have a background in first order logic through the completeness theorem as well as introductory set theory.

The seminar will meet in Room 380-381T Tuesdays 4:30-5:50.

  • Sept 22: Solomon Feferman (Stanford), Introduction to Axiomatic Set Theory, Part I
    Materials: handout
  • Sept 29: Solomon Feferman (Stanford), Introduction to Axiomatic Set Theory, Part II
    Materials: handout
  • Oct 6: Rick Sommer (Stanford), Introduction to Forcing and its Applications
    Abstract (+/-):
    This talk will provide an introduction to Cohen's method of forcing and its application in proving the independence of the Continuum Hypothesis from ZFC. The basic definitions, framework, and ideas will be given without detailed proofs. We will also survey other important independence results that are obtained with the method of forcing.
  • Oct 13: J.T. Chipman (Stanford), Introduction to Classical Descriptive Set Theory
    Materials: handout
    Abstract (+/-):

    Descriptive set theory is part of the mathematical basis for Godel's Program. That is, the program to "decide mathematically interesting questions independent of ZFC in well-justified extensions of ZFC" (Steel). Classical descriptive set theory, however, was developed over 30 years before independence phenomena were even discovered. In this presentation, we describe and motivate the major results of the classical theory, which concern properties of "well-behaved" subsets of the continuum. We then show how, if the classical results are to be developed further, philosophically controversial metamathematical barriers must addressed.

    This presentation is the first of two. The second (Oct 20) concerns principles of definable determinacy. Projective Determinacy, in particular, is thought to solve all of the questions raised by the classical theorists by way of "a well-justified extension of ZFC."

  • Oct 20: J.T. Chipman (Stanford), Principles of Definable Determinacy
    Materials: handout
    Abstract (+/-):

    Suppose that, for every instance of a given class of infinite games, there exists a winning strategy. The sets over which such games are "played" are then called determined. And, if these sets comprise an adequate pointclass, the regularity of this pointclass follows from its determinacy. Martin (1974) proves determinacy of the Borel sets. This result is of foundational significance---Friedman (1971), for instance, supplies a principled sense in which Martin's result is "best possible" in ZFC.

    Last week (Oct 13), by contrast, we saw that the regularity of projective sets beyond those of very low rank is independent of ZFC. If we accept Martin and Steel's proof that the projective sets are determined (1985), however, it follows that the projective sets have the regularity properties after all. This proof is controversial insofar as it assumes the existence of certain large cardinals. This week (Oct 20), we therefore develop some elementary aspects of mathematical and philosophical analyses of that assumption.

  • Oct 27: Larry Moss (Indiana), Thirty Years of Coalgebra: What Have We Learned?
    Materials: handout
    Abstract (+/-):

    Peter Aczel visited CSLI here at Stanford during the year 1984-85. During that time, he wrote the book Non-Well-Founded Sets. The most lasting contribution of that book was not to set theory itself but rather to other areas. These other areas include theoretical computer science, especially those parts of semantics where one needs circular processes and discrete dynamical systems of various sorts. The point is that Aczel phrased his results in the more general language of coalgebras and also worked with some associated concepts from category theory. When people saw that the more general concepts had applications to things in which they were interested, the subject took off. Coalgebra has been extensively developed and now offers a lot of ideas and connections that should be of interest to logicians. To name some examples: generalizations of modal logic in various ways, corecursion, algebraic characterizations of the real unit interval and of some fractal sets, and constructions of reflexive objects in domain theory and of types spaces in economics.

    This talk will survey some of what has been done, centered on constructions of final coalgebras for various functors.

    For background, I suggest the SEP site, especially Section 4.

  • Nov 3: Chris Mierzewski (Stanford), Measurable Cardinals, Constructibility, and Scott’s Theorem
    Materials: handout
    Abstract (+/-):

    In his doctoral thesis, Lebesgue posed his well-known Measure Problem, which asks whether there exists a non-trivial, universal measure over the reals that is translation invariant. Vitali famously settled this question in the negative, relying on the Axiom of Choice.

    A quarter of a century later, Banach formulated a more general question, in a slightly modified form: are there any (uncountable) sets admitting a non-trivial universal measure? This line of inquiry led Ulam to consider the existence of cardinals that admit a special kind of measure with strong additivity properties, which gave rise to the notion of 'measurable cardinal'. Tarski and Ulam then proved that measurable cardinals are inaccessible, which entails that the existence of measurable cardinals cannot be settled within ZFC. Thus emerged, from Lebesgue’s old problem, an interesting set-theoretic question touching upon the foundations of mathematics.

    In 1961 Scott showed, via an elegant ultrapower construction, that the existence of measurable cardinals entails that V is not equal to L. In this sense, the question of whether measurable cardinals exist provides the first example of a large cardinal axiom that discriminates between Godel’s constructible universe and models of ZFC admitting non-constructible sets.

    This talk will be devoted to presenting Scott’s proof. We will see how to generalise the ultrapower construction to proper classes (via "Scott’s trick") and clarify the relationship between measurable cardinals and non-trivial embeddings of the set-theoretic universe.

  • Nov 10: Joseph Helfer (Stanford), The Interpretability Hierarchy
    Abstract (+/-):

    Interpretability formalizes the notion of modelling one first-order theory in another. The relation "S is interpretable in T" forms a partial order on the class of all theories (up to a certain equivalence). This gives us a framework in which we can compare various extensions of ZFC, and in doing so, we discover a surprisingly well-behaved hierarchy of theories.

  • Nov 17: Solomon Feferman (Stanford), The Continuum Hypothesis is Neither a Definite Mathematical Problem Nor a Definite Logical Problem
    Materials: handout
    Abstract (+/-):

    The title of the talk is the same as for the revised version of my contribution to the EFI project, and that is posted on my home page at http://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf/. Participants are urged to read that before the seminar meeting. In my talk I will summarize the arguments and explain technical points. For a quick preview, see also the slides for an earlier version at http://math.stanford.edu/~feferman/papers/CH-Millennium.pdf/.

  • Nov 24: Thanksgiving Break
  • Dec 1: Thomas Icard (Stanford), A Survey on Topological Semantics for Provability Logics
    Abstract (+/-):

    This will be a survey of ideas and results on topological interpretations of provability logics, especially polymodal provability logics. Esakia first proved completeness for the basic Gödel-Löb logic of provability with respect to scattered spaces. Abashidze and Blass (independently) proved completeness w.r.t. a particular scattered space defined on the ordinal ωω. I will discuss older work of my own that extended the Abashidze-Blass result to a polymodal provability logic, which is complete with respect to a polytopological space on ε0. I will then discuss more recent developments, including work by Beklemishev, Fernandez-Duque, Joosten, Gabelaia, Aguilera, and others, generalizing and improving upon much of this.

Winter

The material for the seminar will be drawn from the book, Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena by Jon Barwise and Larry Moss. It is available as a paperback at a reasonable price from online sellers, and also available online at the Stanford Libraries. I will reserve a copy for Tanner Library.

In the first part of the seminar we will cover the material from Part III, Chs. 6-9 on non-wellfounded set theory ZFA. Please read Ch. 2, a review of set theory in the form that will be needed for the construction of a model of ZFA. Participants are hereby encouraged to volunteer in advance for presentations of the chapters in Part III.

The material in the second half of the seminar will be decided later. One natural choice would be to cover Chs. 14-17 with applications to theoretical computer science: largest fixed points, co-induction, co-recursion and co-algebras. But there are several other options to consider.

The seminar will meet in Room 380-381T Tuesdays 4:30-5:50.

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