Workshop on Set Theory and the Philosophy of Mathematics: Titles and Abstracts of Presentations

  • Andreas Blass, Unclear ConceptsAbstract: I plan to discuss several concepts that resist (or at least used to resist) clarification by set-theoretic foundations.  In some cases, I'll make some small suggestions; in others I'll just raise questions or jump to conclusions.
  • John Burgess, Set-Theoretic Foundations and the Structuralist IllusionAbstract: Towards the middle of the last century, set theory as embodied in something like the system ZFC achieved the status of semi-official "foundation" or framework for modern mathematics. Its adoption in the first volume of the Bourbaki encyclopedia was one important event in this process. Roughly speaking, the requirement of rigor today means that it should be possible to view any given piece of mathematics as if it were a chapter in a volume in such an encyclopedia. It should be possible to get to the results in any given paper from set-theoretic first principles together with appropriate definitions. However, the author of a given paper is only responsible for the steps leading from the previous literature to his or her new results. If there are multiple ways to get to the results drawn on from the previous literature, involving different definitions of various notions in terms of set-theoretic primitives, the author of a new piece of mathematics can be indifferent to them. Structuralism in foundations of mathematics, it will be argued, represents a misperception of this fact, distorted by contemporary philosophers' unhealthy preoccupation with so-called ontology.
  • James Cummings, Cardinal arithmetic and set-theoretic methodologyAbstract: I will discuss the history of research in cardinal arithmetic, and what (if anything) we can learn from it about the methodology of set theory.
  • Harvey Friedman, Simple Comprehension Axioms
    Abstract: I show how the formal idea of a simple comprehension axiom (and axiom scheme) generates, exactly, various fragments of ZFC including ZF with weakened foundation. These kinds of results suggest a formal explanation for our adoption of the usual axiom systems, that provides an alternative to the traditional realist explanation. We conjecture that all of the main formal systems of f.o.m. are exactly generated by simple syntactic principles, including extensions of ZFC via basic large cardinal hypotheses.
  • Akihiro Kanamori, The Axiom of Replacement: A Larger Mathematical and Historical
    Perspective

    Abstract: A larger mathematical and historical perspective is put on
    the Axiom of Replacement, thereby to affirm its importance as a central
    axiom of set theory.
  • Juliette Kennedy, On Formalism Freeness
    Abstract: We propose a robustness test for notions of definability in set theory, expanding upon some of Gödel's remarks on the topic from his 1946 Princeton Bicentennial lecture.
  • Peter Koellner, Indeterminateness in Set Theory
    Abstract: There is widespread disagreement on the question of whether
    certain (local) statements of set theory (most notably the
    Continuum hypothesis) are ``determinate''.  On the positive side
    some have maintained that the categoricity results ensure that
    such statements are determinate.  On the negative side some have
    maintained that independence (or, more precisely, independence +
    X (for various X)) ensures that such statements are
    indeterminate.

    The talk has two parts.  In the first part I will survey the
    standard arguments for and against determinateness in set theory
    and I will argue that each is problematic.  In the second part I
    will address the question of what it would take to settle the
    issue, in particular, what it would take to have a strong case
    for the claim that some such statement is indeterminate.  I will
    not be able to answer that question but I will map out some
    scenarios that may illuminate what is at issue.
  • Donald A. Martin, Philosophical issues about the hierarchy of sets
    Abstract: I will discuss some philosophical questions about the cumulative
    hierarchy of sets, its levels, and their theories. Some examples:

    (1) It is sometimes asserted one cannot quantify over everything. A
    related assertion is that each of our statements about the
    universe of sets can from a different perspective be seen as a
    statement about some Va.  Thus the class-set distinction
    is really a relative one. Does this make sense? Is it right?

    (2) Is the first order theory of V determinate?  Does every sentence
    have a truth value?  Are there levels of the hierarchy whose first
    order theories are indeterminate? If so, what is the lowest such
    level? What about L and the constructibility hierarchy?

    (3) There are many examples of proofs of a statement about one level of
    the hierachy that use principles about a higher level. Under what
    conditions and in what sense do these count as establishing the
    lower level statement.

    I will discuss these questions mainly from a viewpoint that takes
    mathmematics to be about basic mathematical concepts, e.g., those of
    natural number, real number, and set.
  • Justin Moore, Π2 maximality and the Continuum Hypothesis
    Abstract: It is known that there is a strongest consistent forcing
    axiom, namely the forcing axiom for partial orders which preserve
    stationary subsets of $\omega_1$ (also known as Martin's Maximum or MM).
    The consistency of this axiom was established by Foreman, Magidor, and
    Shelah relative to the existence of a supercompact cardinal. Since MM
    has proved very effective at settling statements left independent of
    ZFC, it is natural to ask whether there is an analogous optimal forcing
    axiom which is relatively consistent with the Continuum Hypothesis. One
    precise way to cast this question is whether there are two $\Pi_2
    $-sentences in the language of $(H(\omega_2), \in , NS_{\omega_1})$
    which are each $\Omega$-consistent with CH but which jointly negate CH.
    This problem is due to Woodin, who showed that the answer is negative if
    one replaces CH with ZFC. We show that the answer to Woodin's problem is
    positive and in the process establish that there are two preservation
    theorems for not adding reals which can not be subsumed into a single
    iteration theorem. This is joint work with David Aspero and Paul Larson.
  • W. Hugh Woodin, Ultimate L
    Abstract: I shall survey the current status of the search for "ultimate-L"--which is the ultimate enlargement of Gödel's constructible universe L.