The following speakers are confirmed:
The Consistency of a Club-Guessing Failure at the Successor of a Regular Cardinal. Abstract / Slides
Self-Reference in Arithmetic. Abstract / Preprint
An Inconsistency in the Large Cardinal Hierarchy? Abstract
Felix Hausdorff and the Foundations of Mathematics. Abstract
Probabilistic Theories of Type-Free Truth and Probability. Abstract / Slides
The Set Theory of Generalized Logics. Abstract / Slides
Gödel's Program. Abstract / Slides
Generic Absoluteness, Determinacy, and the Form of Ultimate L. Abstract / Slides |
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The Conference is partly supported by the Heibronn Institute as a Heilbronn Day, by the British Logic Colloquium, by the Bristol Centre for Science & Philosophy, and by the School of Mathematics.
Set theorists have discovered many mutually incompatible natural theories extending ZFC. It is possible that these incompatibilities will be resolved by interpreting all such theories in a useful common framework theory.
I will give a survey of recent results and theories concerning (i) probability measures for sentences that can speak about their own truth or falsity, and (ii) probability measures for sentences that can speak about their own probability. As I am going to show, one of the crucial issues in that area of research is sigma-additivity: does it have to be given up for type-free probability?
The hierarchy of large cardinals provides us with a canonical means to climb the hierarchy of consistency strength. There have been any purported inconsistency proofs of various large cardinal axioms. For example, there have been many proofs purporting to show that measurable cardinals are inconsistent. But to date the only proofs that have stood the test of time are those which are rather transparent and simple, the most notable example being Kunen's proof showing that Reinhardt cardinals are inconsistent. The Kunen result, however, makes use of AC. And long standing open question is whether Reinhardt cardinals are consistent in the context of ZF.
In this talk I will survey the simple inconsistency proofs and then raise the question of whether perhaps the large cardinal hierarchy outstrips AC, passing through Reinhardt cardinals and reaching far beyond. There are two main motivations for this investigation. First, it is of interest in its own right to determine whether the hierarchy of consistency strength outstrips AC. Perhaps there is an entire "choicless" large cardinal hierarchy, one which reaches new consistency strengths and has fruitful applications. Second, since the task of proving an inconsistency result becomes easier as one strengthens the hypothesis, in the search for a deep inconsistency it is reasonable to start with outlandishly strong large cardinal assumptions and then work ones way down. This will lead to the formulation of large cardinal axioms (in the context of ZF) that start at the level of a Reinhardt cardinal and pass upward through Berkeley cardinals (due to Woodin) and far beyond. Bagaria, Woodin, and myself have been charting out this new hierarchy. I will discuss what we have found so far.
Exactly 100 years ago, after a long phase of foundational uncertainty Felix Hausdorff's "Grundzüge der Mengenlehre" (Foundationas of Set Theory) established set theory as a comprehensive field of mathematics. Hausdorff advocated set theory as a universal foundation of mathematics. He followed David Hilbert's axiomatic method and formalism. Hausdorff's position corresponds closely to the anti-metaphysical stance in his philosophical book "Das Chaos in Kosmischer Auslese" (Chaos in Cosmic Selection). A formalist like Hausdorff selects consistent axiom systems from the chaos of mathematical possibilities, guided by various criteria. Not least by intellectual and aesthetics criteria.
The current scenarios forecast the construction of two flavors of Ultimate L with very different properties. We discuss emerging evidence that there is in fact just one flavor.
A generalized logic is a mechanism of extending first order logic in order to express properties of mathematical structures or of elements of such structures. A prime example is higher order logic, like second order logic. A generalized logic is typically sensitive to the set theoretical frame work in which the mathematical structure is embedded. This creates an interesting interplay between properties of the logic, like compactness, Skolem-Lowenheim theorems etc. and the underlying Set Theory.
The interaction can go both ways: A desired properties of the logic under consideration can be used as a motivation for new axioms for Set Theory, for new notions of large cardinals, etc. On the other hand analysis of the possible properties of a given logic in different set theoretic universes can give some insight into the the strength of the logic and its relations with other logics.For instance a typical (admittedly vague) problem is the extent by which a given logic is really logic, or is it Set Theory in disguise.
In this talk we shall show some examples of such interplay .