# Symposia on the foundation of Mathematics, July 7-8, 2014

The conference has the following general aims and objectives:

1. To bring to a wider philosophical audience the different approaches that one can take to the set-theoretic foundations of mathematics.

2. To elucidate the pressing issues of meaning and truth that turn on these different approaches.

3. To address philosophical questions concerning the need for a foundation of mathematics, and whether or not set theory can provide the necessary foundation

Date and Venue: 7-8 July 2014 – Kurt Gödel Research Center, Vienna

Confirmed Speakers:

• Sy-David Friedman (Kurt Gödel Research Center for Mathematical Logic),
• Hannes Leitgeb (Munich Center for Mathematical Philosophy)

Programme:

DAY 1
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.’
1300-1500 Lunch
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.’

DAY 2
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.’
1300-1500 Lunch
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.’

Call for Papers: We welcome submissions from scholars (in particular, young scholars, i.e. early career researchers or post-graduate students) on any area of the foundations of mathematics (broadly construed). Particularly desired are submissions that address the role of set theory in the foundations of mathematics, or the foundations of set theory (universe/multiverse dichotomy, new axioms, etc.) and related ontological and epistemological issues. Applicants should prepare an extended abstract (maximum 1’500 words) for blind review, and send it to sotfom [at] gmail [dot] com. The successful applicants will be invited to give a talk at the conference and will be refunded the cost of accommodation in Vienna for two days (7-8 July).

Submission Deadline: 31 March 2014

Notification of Acceptance: 30 April 2014

Scientific Committee: Philip Welch (University of Bristol), Sy-David Friedman (Kurt Gödel Research Center), Ian Rumfitt (University of Birmigham), John Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Gödel Research Center), Neil Barton (Birkbeck College), Chris Scambler (Birkbeck College), Jonathan Payne (Institute of Philosophy), Andrea Sereni (Università Vita-Salute S. Raffaele), Giorgio Venturi (Université de Paris VII, “Denis Diderot” – Scuola Normale Superiore)

Organisers: Sy-David Friedman (Kurt Gödel Research Center), John Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Gödel Research Center), Neil Barton (Birkbeck College), Carolin Antos (Kurt Gödel Research Center)

# Talks and Short Abstracts

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’

TBA

Jose Ferreirós: Issues of evidence in set theory: some comparative remarks.’

TBA

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.’

TBA

Brice Halimi: Models as Universes.’

Georg Kreisel raised the problem as to whether any logical consequence of ZFC is true. Kreisel and George Boolos both proposed an answer, taking “truth” to mean truth in the background set-theoretic universe. This talk advocates another answer, which lies at the level of models of set theory. Firstly, after analyzing Kreisel’s and Boolos’ solutions, I will propose one way in which any model of set theory can be compared to a background universe and shown to contain “internal models.” Logical consequence w.r.t. any given model of ZFC can thus be defined. I will then present results bearing on internal models and their implications for Kreisel’s problem. Finally, taking internal models as accessible worlds, I will introduce an “internal modal logic” in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4.

Chris Scambler: Absoluteness and Indeterminacy: On the Philosophical Signiﬁcance of Semi-Constructive Set Theory.’

Where systems of set theory employing mixed intuitionistic and classical logics have been studied, e.g. in the work of Tharpe[1971], Pozgay[1972], Feferman[2011] and Rathjen[2014], they have always been motivated by metaphysical arguments concerning the “unfinished”, “potential” or “indefinite” character of the set-theoretic universe. However, the distinction between definite and indefinite often comes down to a matter of taste (witness arguments concerning the definiteness or otherwise of the powerset of omega), and even where what’s indefinite is agreed on the connection between indefinite extensibility of a given domain and the putative inapplicability of classical reasoning over it is somewhat obscure. In this paper, I argue that such metaphysical arguments are not the best available, and seek to offer an alternative. To that end, I present a semantic motivation for a particular system of set theory employing a mixed logic due to Solomon Feferman, known as Semi Constructive Set theory (SCS). I show how the axioms of SCS can be viewed as motivated by the desire to restrict classical logic to set-theoretic assertions that are, in a manner to be made precise, determinate in sense. I explore the extent of the indeterminacy in set theory diagnosed by such arguments, showing the boundary to be much more precise than it was on the metaphysical approach. I also consider the consequences acceptance of such a view might have for our understanding of ZFC as normally practised.

Hans-Christoph Kotzsch: Homotopy Type Theory and Foundations of Mathematics.’

TBA

Walter Dean: On reflection, Proof- and Set-Theoretic.’

TBA

Johannes Korbmacher and Georg Schiemer: On Structural Properties.’

Informally, structural properties are usually characterized in one of two ways: either as the properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects.
In this talk we present two formal explications of structural properties, corresponding to these two informal characterizations. We wish to reach two goals: First, we wish to get clear on how the two accounts capture the intuition that structural properties are “grounded in structure”. Second, we wish to understand the relation between the two explications of mathematical properties. As we will show, the  two characterizations do not determine the same class of properties. From this observation we draw some philosophical conclusions about the possibility “correct” analysis of structural properties.