Month: June 2014

Symposium I, KGRC 7-8 July: Talks and Short Abstracts


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.’

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.’


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.

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. 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.’

In the talk we address the question as to the potential significance of Homotopy Type Theory (HoTT) as a foundation for mathematics from a philosophical perspective. In particular, what are the differences to set-theoretic foundations, both mathematically and conceptually? Mathematically, HoTT extends other type-theoretic approaches, being based on Martin-Löf dependent type theory. With this comes a conceptual connection with categorically-minded `structural’ set theories as studied in categorical logic or algebraic set theory. Thus, in a way, some arguments in favour of HoTT carry over from discussions about the merits of structural set theories; such as for instance to take the notion of function as primitive.

On the other hand, HoTT owns more genuine aspects mainly connected with the treatment of identity types and the univalence axiom. Awodey in a recent paper explained how the resulting type theory is particularly suited to fit structuralists’ needs in that identity and isomorphism – more generally, equivalence – in a precise technical sense coincide. Extending these ideas we argue that in many concrete circumstances the notion of identity in HoTT coincides with notions of structural sameness between structures that is usually viewed as appropriate in set-based mathematics. In other words, the HoTT-notion of equivalence between types (i.e. structures) in general coincides with certain intuitive notions of sameness of structures, internalized in HoTT by definitions very similar to the usual ones; e.g. isomorphism of sets or equivalence of categories.

This a non-trivial assertion but also is not a general statement subject to proof, since in each case the proposed match depends on how to rephrase the usual definitions. By exhibiting various examples we try to argue, however, that the notion of identity in HoTT may be particularly suited so as to subsume the intuitive notions of structural sameness in many instances.

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

Proof theoretic reflection principles are statements or schema which express the soundness of a theory within its own language. Set theoretic reflection principles are statements or schema which express that any property of the set theoretic universe V already holds of
some set within V.  Such principles have been studied separately since the 1960s, respectively beginning with the work of Kreisel and Feferman in proof theory, and Montague, Levy, and Bernays in set theory. Several subsequent authors (e.g. Kreisel & Levy, Takeuti, and
Reinhardt) have also hinted at possible connections between them — both conceptual and mathematical.  The goal of this talk will be to critically explore such analogies in light of results connecting proof
theoretic reflection to transfinite induction and set theoretic reflection to (small) large cardinals.

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.