Session 1 – Chair: Carolin Antos
1000-1005 Introductory Remarks
1005-1135 Sy-David Friedman: `The Hyperuniverse Programme.’
1135-1145 Coffee Break
1145-1300 Jose Ferreirós: `Issues of evidence in set theory: some comparative remarks.’
Session 2 – Chair: John Wigglesworth
1500-1615 Zeynep Soysal: `What is the Universe of Sets?’
1615-1730 Toby Meadows: `The Generic Multiverse Debate.’
1730-1745 Coffee Break
1745-1900 Brice Halimi: `Models as Universes.’
Session 1- Chair: Claudio Ternullo
1000-1130 Hannes Leitgeb: `On Mathematical Structuralism.’
1130-1145 Coffee Break
1145-1300 Chris Scambler: `Absoluteness and Indeterminacy: On the Philosophical Signiﬁcance of Semi-Constructive Set Theory.’
Session 2 – Chair: Neil Barton
1500-1615 Hans-Christoph Kotzsch: `Homotopy Type Theory and Foundations of Mathematics.’
1615-1730 Walter Dean: `On reflection, Proof- and Set-Theoretic.’
1730-1745 Coffee Break
1745-1900 Johannes Korbmacher and Georg Schiemer: `On Structural Properties.’
Sy-David Friedman: `The Hyperuniverse Programme.’
There are both extrinsic and intrinsic reasons for accepting the axioms of ZFC as true assertions about sets. However these axioms leave many interesting questions of set theory, such as the Continuum Hypothesis (CH) and the existence of large cardinals unanswered. Although there is considerable extrinsic evidence coming from set-theoretic practice for a number of axiom-candidates beyond ZFC, no such extrinsic evidence is sufficient to convincingly resolve CH or the existence of large cardinals. Moreover, there has been no adequate study of extrinsic evidence for new axioms of set theory coming from other areas of logic or mathematics.
The aim of the Hyperuniverse Programme is to provide a new source of evidence for axioms of set theory based on intrinsic properties of possible universes of sets. These “pictures of V” are taken to be elements of the Hyperuniverse, the collection of countable transitive models of the ZFC axioms. The principles of “maximality” and “omniscience” are formulated precisely as mathematical criteria for the preference of certain universes, and first-order properties shared by these preferred universes are proposed as intrinsically-based candidates for new axioms of set theory.
In this talk I will review the current state of the programme. Preliminary results suggest that new axioms of set theory which are both intrinsically-based and compatible with set-theoretic practice may include statements asserting the existence of weakly compact cardinals, the existence of inner models with measurable cardinals as well as strong failures of CH.
Hannes Leitgeb: `On Mathematical Structuralism’
There are different versions of structuralism in present-day philosophy of mathematics which all take as their starting point the structural turn that mathematics took in the last two centuries. In this talk, I will make one variant of structuralism—ante rem structuralism—precise in terms of an axiomatic theory of unlabeled graphs as ante rem structures. And then I will use that axiomatic theory in order to address some of the standard objections to ante rem structuralism that one can find in the literature. Along the way, I will discuss also other versions of mathematical structuralism, and I will say something on how the emerging theory of ante rem structures relates to modern set theory.
José Ferreirós: `Issues of evidence in set theory: some comparative remarks.’
The talk will discuss what is probably the greatest success story in set theory of the late 20th century, namely the solution to questions in descriptive set theory (DST) by the axiom of Projective Determinacy (and its derivation from large cardinals). Our question is about the evidence for this proposed axiom, where as usual (Maddy, Martin, Steel) we shall distinguish between intrinsic and extrinsic evidence. But beyond that, and trying to clarify the situation, we shall distinguish between the kinds of evidence that have been offered for different axioms or candidate axioms in set theory. Some comparative remarks about the aims and methods of DST in its classical period (first half of 20th cent.) and its more contemporary development will help us make that distinction. The outcome will be a somewhat skeptical assessment of the current state of the matter.
Zeynep Soysal: `What is the Universe of Sets?’
According to the iterative conception of sets, there is no set of all sets. Two questions arise: (1) what—if not a set—is the universe of all sets, and (2) why is it not a set? The actualist’s response to (1) is that the universe of sets is a “completed totality;” her response to (2) is that the universe of sets is “too large” to be a set. The potentialist’s response to (1) is that the universe of sets is “merely potential;” her response to (2) is that since the universe is merely potential, there is no actually existing plurality of all sets, hence also no set of all sets.
I argue that neither the actualist nor the potentialist answer to (2) is satisfactory. In doing so, I diagnose a problem common to actualists and potentialists: they mistakenly seek a metaphysical explanation of why the universe of sets is not a set—an explanation that appeals to the metaphysical nature of the universe of sets. I conclude by outlining a different approach. It consists of four claims: (i) the fact that the universe of sets is not a set is a conceptual truth; (ii) this provides an adequate answer to (2); (iii) the answer to (1), on the other hand, is left open by the concept of set and the choice between actualism and potentialism concerning the metaphysical nature of the universe of sets is a matter of expedience; (iv) actualism fares better than potentialism with respect to expedience.
Toby Meadows: `The Generic Multiverse Debate.’
This paper is about the practice and methodology of set theory at its cutting edge. While set theory is certainly a technical discipline, it possesses a number of features which are not shared by similarly technical fields. My goal is to draw out some of these features in order to make clearer what some leading set theorists are doing. The setting for this investigation will be a debate between Hugh Woodin and John Steel regarding the subject matter of set theory itself. In brief, Woodin defends the traditional view that there is a unique cumulative hierarchy of sets, so to speak, out there and that this is what set theorists study. We’ll call this the universe view. Steel, on the other hand, takes the view that our intuition of the cumulative hierarchy is illusory and that we would be better served by thinking of set theory as focused on a plurality of set theoretic universes linked by forcing. We’ll call this the (generic) multiverse view.
The ultimate goal of the paper will be to illustrate that this debate is in a kind of philosophical stalemate wherein neither side is in a strong position to persuade the other to their cause.