Below you can find a conference programme. There’s also the short abstracts to whet your appetite! We look forward to seeing you in Munich!

Programme:

Day 1 – Monday, October 9, 2017, LMU Main Building, room A 022

10:00-10:15 Welcome

10:15-11:45 Stephen G. Simpson `Foundations of mathematics: an optimistic message’

11:45-12:15 Coffee break

12:15-13:15 Sam Sanders `Two is enough for chaos in reverse mathematics’

13:15-15:00 Lunch break

15:00-16:00 Michał Tomasz Godziszewski `What do we need to prove that satisfaction is not absolute? Generalization of the Nonabsoluteness of Satisfaction Theorem’

16:00-16:30 Coffee break

16:30-18:00 Walter Dean `Basis theorems and mathematical knowledge de re and de dicto’

19:00 Conference dinner

Day 2 – Tuesday, October 10, 2017, Main Building, room E 006

10:15-11:45 Benedict Eastaugh `On the significance of reverse mathematics’

11:45-12:15 Coffee break

12:15-13:15 Marta Fiori Carones `Interval graphs and reverse mathematics’

13:15-15:00 Lunch break

15:00-16:00 Eric P. Astor `Divisions in the reverse math zoo, and the weakness of typicality’

16:00-16:30 Coffee break

16:30-18:00 Marianna Antonutti Marfori `De re and de dicto knowledge in mathematics: some case studies from reverse mathematics’

Day 3 – Wednesday, October 11, 2017, LMU Main Building, room A 022

10:15-11:45 Takako Nemoto `Finite sets and infinite sets in constructive reverse mathematics’

11:45-12:15 Coffee break

12:15-13:15 Vasco Brattka `Weihrauch complexity: Choice as a unifying principle’

13:15-15:00 Lunch break

15:00-16:00 Alberto Marcone `Around ATR_0 in the Weihrauch lattice’

16:00-16:30 Coffee break

16:30-18:00 Marcia Groszek `Reverse recursion theory’

Abstracts:

Stephen G. Simpson `Foundations of mathematics: an optimistic message’

Historically, mathematics has often been regarded as a role model

for all of science — a paragon of abstraction, logical precision,

and objectivity. The 19th and early 20th centuries saw tremendous

progress. The great mathematician David Hilbert proposed a sweeping

program whereby the entire panorama of higher mathematical

abstractions would be justified objectively and logically, in terms

of finite processes. But then in 1931 the great logician Kurt

Gödel published his famous incompleteness theorems, leading to an

era of confusion and skepticism. In this talk I show how modern

foundational research has opened a new path toward objectivity and

optimism in mathematics.

Sam Sanders `Two is enough for chaos in reverse mathematics’

Reverse Mathematics studies mathematics via countable approximations, also called codes. Indeed, subsystems of second-order arithmetic are used, although the original Hilbert-Bernays system H includes higher-order objects. It is then a natural question if anything is lost by the restriction to the countable imposed by second-order arithmetic. We show that this restriction fundamentally distorts mathematics. To this end, we exhibit ten theorems from ordinary mathematics which involve type two objects and cannot be proved in any (higher-type version) of Pi_k^1-comprehension, for any k. Said theorems are however provable in full (higher-order) second-order arithmetic and intuitionism, as well as often in (constructive) recursive mathematics.

Michał Tomasz Godziszewski `What do we need to prove that satisfaction is not absolute? Generalization of the Nonabsoluteness of Satisfaction Theorem’

We prove the following strengthening of the Hamkins-Yang theorem on the nonabsoluteness of satisfaction:

For any theory interpreting PA and defining the arithmetical truth and for any countable -nonstandard model there is an isomorphic model such that (the natural numbers of the models agree), but i.e. the true arithemetics of the models are incompatible

We will provide a philosophical discussion of the theorem in the context of the debate on the so-called multiverse perspective in foundations of mathematics.

There is yet a question of reverse-mathematical strength of the presented results. If one assumes that is recursively saturated, then even is enough. Without this assumption, however, we need to use the result saying that in the countable recursive saturation implies chronic resplendency, and the reverse-mathematical status of this implication is (according to our knowledge) an open question. Further, if we phrase the assumptions of the result in a way that instead of postulating recursive saturation we require that an arithmetical truth predicate in the model satisfies the theory axiomatizing the concept of full satisfaction class, then to get some of our corollaries we need to appeal to Lachlan’s theorem saying that admitting a satisfaction class implies recursive saturation. As above, the reverse-mathematical status of this implication is an open problem. We therefore leave some questions concerning strength not only of the theory from above, but reverse-mathematical status of the results on the nonabsoluteness of satisfaction themselves.

Walter Dean `Basis theorems and mathematical knowledge de re and de dicto’

TBA

Marta Fiori Carones `Interval graphs and reverse mathematics’

This paper deals with the characterization of interval graphs from the point of view of reverse mathematics, in particular we try to understand which axioms are needed in the context of subsystems of second order arithmetic to prove various characterizations of these graphs. Furthermore, we attempt to analyze the interplay between interval orders and interval graphs. Alberto Marcone (2007) already settled the former question for interval orders and this paper follows his path. An interval graph is a graph (V,E) whose vertices can be mapped in intervals of a linear order L such that if two vertices are related, then the intervals associated to them overlap. Different notions of intervals give raise to different notions of interval graphs (and orders), which collapse to one in WKL0. On this respect, interval graphs show the same behavior of interval orders. On the other hand, interval graphs and interval orders have different strength with respect to their characterizations in terms of structural properties. While RCA0 suffices to prove the combinatorial characterization of interval orders, WKL0 is required for interval graph s. Moreover, given an interval graph it is possible to define an associated interval order and vice versa. Even in this respect, the different definitions of interval graphs and orders mentioned before play a role in analysing the strength of this theorems.

Eric P. Astor `Divisions in the reverse math zoo, and the weakness of typicality’

In modern reverse math, we have begun to discover some exceptional subsystems that do not fall into the linearly-ordered Big Five, particularly concentrated between ACA0 and RCA0. These exceptional systems form a structure that is sometimes called the reverse-mathematics Zoo. Between ACA0 and RCA0, the Zoo can be seen to divide into three branches: roughly, combinatorics, randomness, and genericity. (Prominent examples on each branch include RT22, WWKL, and Pi^0_1G respectively.) We raise questions about the significance of this division in the Zoo, and obtain the first theorem describing this large-scale structurer. Noting that both randomness and genericity are notions of typicality, we find a strict limitation on the strength of principles stating that sufficiently typical sets exist; using this, for nearly all known principles P in the Zoo, we determine whether P follows from any randomness- or genericity-existence principle.

Marianna Antonutti Marfori `De re and de dicto knowledge in mathematics: some case studies from reverse mathematics’

TBA

Takako Nemoto `Finite sets and infinite sets in constructive reverse mathematics’

We consider, for a set A of natural numbers, the following notions of finiteness

FIN1: There are k and m_0,,,,m_{k-1} such that A={m_0,…,m_{k-1}};

FIN2: There is an upper bound for A;

FIN3: There is m such that for all B\subseteq A(|B|<m);

FIN4: It is not the case that, for all x, there is y such that y\in A;

FIN5: It is not the case that, forall m, there is B\subseteq A such

that |B|=m,

and infiniteness

INF1: There are no k and m_0,…,m_{k-1} such that A={m_0,…,m_{k-1}};

INF2: There is no upper bound for A;

INF3: There is no m such that for all B\subseteq A(|B|<m);

INF4: For all y, there is x>y such that x\in A;

INF5: Forall m, there is B\subseteq A such that (|B|=m).

We systematically compare them in the method of constructive reverse mathematics.

We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, including the axiom called bounded comprehension.

Vasco Brattka `Weihrauch complexity: Choice as a unifying principle’

TBA

Alberto Marcone `Around ATR_0 in the Weihrauch lattice’

The classification of mathematical problems in the Weihrauch lattice is a line of research that blossomed in the last few years. So far this approach has mainly dealt with statements which are provable in ACA0 and below. On the other hand the study of multi-valued functions arising from statements laying at higher levels (such as ATR0) of the reverse mathematics spectrum is still in its infancy. We pursue this study by looking at multi-valued functions arising from statements such as the perfect tree theorem, comparability of well-orders, \Delta^0_1 and \Sigma^0_1-determinacy, \Sigma^1_1-separation, weak \Sigma^1_1-comprehension, \Delta^1_1 and \Sigma^1_1-comprehension. As usual, choice functions provide significant milestones to locate these multi-valued functions in the Weihrauch lattice.

Marcia Groszek `Reverse recursion theory’

TBA

]]>

We’re very grateful to the DFG, LMU, and MCMP in this regard and for making the conference possible. It promises to be a great meeting!

]]>Reverse mathematics is concerned with examining exactly which axioms are necessary for various central mathematical theorems and results. The program is a relatively new one in the foundations of mathematics. Its basic goal is to assess the relative logical strengths of theorems from ordinary (non-set-theoretic) mathematics. To this end, for a given mathematical theorem T, one tries to find the minimal natural axiom system that is capable of proving T. In logical terms, finding the minimal axiom system equates to finding a collections of axioms such that each axiom follows from T (assuming a weak base system of axioms). In doing so, one shows that each axiom is necessary for T to hold. Because, by hypothesis, T follows from the axioms as well, the goal of reverse mathematics is to find axiom systems to which the theorems of ordinary mathematics are equivalent. It turns out that most theorems are equivalent to one of five subsystems of second order arithmetic.

The main objective of the conference is to explore the philosophical significance of reverse mathematics as a research program in the foundations of mathematics. The event will provide a forum for experts and early career researchers to exchange ideas and develop connections between philosophical and mathematical research in reverse mathematics. Specifically, the following research questions will be addressed:

1. How are philosophical debates informed by divisions between the relevant five subsystems of second order arithmetic, e.g., the debate between predicativism and impredicativism?

2. How should we understand the divisions between these five systems in terms of any natural distinctions they map on to?

3. How exhaustive are these five systems, especially in the sense of how they map onto natural divisions?

4. How does reverse mathematics relate to and inform our understanding of more traditional foundations of mathematics like ZFC, e.g., concerning the existence of large cardinals?

Confirmed Speakers:

Marianna Antonutti Marfori (Munich Center for Mathematical Philosophy, LMU Munich)

Walter Dean (University of Warwick)

Benedict Eastaugh (University of Bristol)

Marcia Groszek (Dartmouth College)

Takako Nemoto (Japan Advanced Institute of Science and Technology)

Stephen G. Simpson (Pennsylvania State University and Vanderbilt University)

Call for Abstracts:

We invite the submission of abstracts, suitable for a 40 minute talk, on topics related to any aspects of reverse mathematics. We encourage submissions from early-career researchers and PhD students. Please send an abstract of around 1000-1500 words by email to sotfom@gmail.com in PDF format. Abstracts should be prepared for blind review. The author’s name, paper title, institutional affiliation, and contact details should be included in the email.

Dates and Deadlines:

Submission deadline: 6 August 2017

Notification of acceptance: 15 August 2017

Registration deadline: 1 October, 2017

Conference: 9 – 11 October, 2017

For further details on the conference, please visit: https://sotfom.wordpress.com

Organisers:

Carolin Antos-Kuby (University of Konstanz), Neil Barton (Kurt Gödel Research Center, Vienna), Lavinia Picollo (Munich Center for Mathematical Philosophy), Claudio Ternullo (Kurt Gödel Research Center, Vienna), John Wigglesworth (University of Vienna)

SotFoM4: Reverse Mathematics is generously supported by the Munich Center for Mathematical Philosophy, LMU Munich, and the Deutsche Forschungsgemeinschaft.

]]>The purpose of the conference was to bring together scholars who, in the last years, have contributed to the ongoing debate on the foundations of set theory, in particular on such topics as the universe/multiverse dichotomy, new axioms of set theory and their ontological and epistemological features, different forms of justification for the acceptance of new axioms, competing foundations and, finally, the Hyperuniverse Programme (HP), which is currently investigated at the KGRC by S. Friedman and collaborators.

The conference was opened by Tatiana Arrigoni’s talk, which aimed to assess the current status of the HP, as recently developed by Friedman, Antos, Honzik and Ternullo. Arrigoni acknowledged that much work has been done within the programme in the direction of connecting multiverse axioms to the concept of set, but she also pointed out that further work is needed to make a full case for the `intrinsicness’ of the programme’s maximality principles. She also encouraged further work on the issue of why the programme believes that intrinsically justified axioms are particularly valuable for the fruitful (and correct) development of set theory.

In his talk, Giorgio Venturi explained how forcing may lend support to a realist conception of set theory, by using what he calls trans-universe realism, whose plausibility, in turn, is grounded on some recent mathematical results due to Hamkins. Pushing this interpretation of forcing even further, Venturi suggested that the notion of arbitrary set, which is an integral part of what he sees as a feature of a realist attitude in set theory (Bernays’ quasi-combinatorialism), may find a sharper formulation in that of generic set, as used in forcing.

Finally, in the last talk in the morning of Day 1, Dan Waxman assessed the argument that the independence phenomenon in set theory leads one to viewing some set-theoretic statements as indeterminate. The specific argument assessed by Waxman takes determinacy to arise either from the `world’ or from our `practice’. Now, if one accepts what Waxman defines, respectively, a metaphysical (1) and a cognitive (2) constraints on our theories, whereby, respectively, (1) objects are not ineliminable in our best account of mathematics and (2) we cannot attribute non-mechanical powers to our minds, then there will, of necessity, be indeterminate statements in set theory.

In the afternoon, Matteo Viale reviewed some of the reasons why forcing axioms have proved to be `successful’ in set theory. Among others, he pointed to the following two: (1) they are equivalent to key, well-established principles, such as the Axiom of Choice, and (2) they lead to absoluteness results within the set-generic multiverse.

In the last talk of Day 1, Øystein Linnebo presented his well-known modal version of the axioms of set theory using plural quantification, showing how it fits the requirements of a potentialist conception which originates from Cantor’s work (which famously identifies proper classes such as the ordinals as inconsistent, qua incompletable, multiplicities). In Linnebo’s account, this means, in particular, that (1) not all objects in V are given immediately (rather, they are produced gradually) and (2) that truths are `created’ as the hierarchy gradually unfolds.

On Day 2, Mary Leng reviewed Penelope Maddy’s recent `Defending the New Axioms’ book, where Maddy advocates a form of realism labelled `thin’ realism or `arealism’. In particular, Leng described how, by this conception, the old dispute about the existence of mathematical objects is irrelevant, while it is still relevant for it, and it still makes sense to ask, whether CH, e.g., is true or false. She then took a step further, by pointing out that arealism might, in fact, be compatible with the view that there is no fact of the matter about whether CH is true or false.

Afterwards, Neil Barton showed that the HP may be compatible with several pictures of the universe of sets and, in particular, with that arising from an `absolutist’ conception of V. The bulk of Barton’s strategy consists in showing that not only forcing extensions of V can be coded into models definable in V itself, as already shown by Hamkins, but, in general, that this, using V-logic, applies to all outer models of the Morse-Kelley axioms (which, in turn, best express the absolutist viewpoint). V-logic is a basic ingredient to express HP’s maximality principles and, thus, Barton’s project may open the way to an `absolutist’ construal of the HP.

The rest of Day 2 was spent making a trip in the Wienerwald. After having lunch in the rural Mostalm, Dr Peter Telec kindly offered to lead a relaxing walk through the beautiful Viennese woods.

Day 3 opened with Geoffrey Hellman’s talk detailing a height-potentialist conception of V. Hellman showed us how his account, relying on a modal version of Zermelo’s second-order ZFC, is adequate to express some crucial intuitions behind set theory, including the \emph{set/class} distinction as proposed by Zermelo, as well as the indefinite extendability of the universe. Moreover, he showed how his account could also more naturally justify second-order reflection, which would take us beyond the inaccessibles and would help justify all small large cardinals.

Talking after Hellman, Sam Sanders showed that, using non-standard analysis, one can successfully respond to Voevodsky’s recently questioning the adequacy of ZFC as a foundation of mathematics, insofar as the latter cannot be `computational’. This presupposition led Voevodsky to reject ZFC as a foundation and advocate HoTT (Homotopy Type Theory) as a plausible alternative. As shown by Sanders, `computational’, here, does not mean `implemented by a computer’, but rather `constructive’ in the sense of Per-Löf’s intuitionistic type-theoretic system.

Subsequently, Emil Weydert explored the intiriguing topic of how the HP could be used to produce an axiom induction framework. The basic ingredient is given by formalising `multiverse reasoning’ in terms of non-monotonic reasoning, whereby the addition of new hypotheses (i.e., new axioms) leads to finding differring conclusions. Weydert’s project also envisages taking into account other parameters which are relevant to selecting new axioms.

In his talk, Douglas Blue started with a definition of maximality in set theory as the demand that a candidate axiom maximise the interpretative power of a theory T. But he showed us the interesting case of the axiom , which satisfies the aforementioned definition (that is, as shown by Woodin, it maximises the theory of as far as -sentences are concerned), but is in contrast with a more intuitive version of maximality, one entailing that the power-set operation adds novel structure: if we adopt , then we have that is bi-interpretable with and, thus, fails to satisfy the second version of maximality.

Our last speaker was Sy Friedman, who described different notions of `new axiom’ and justifications thereof. Friedman reviewed the maximality principles which have been investigated within the HP and, arguably, related to the concept of set (and, thus, intrinsically justified). Now, one further criterion to judge the value of an axiom is to see whether it is useful. However, here Friedman departs from the standard interpretation of this, by suggesting that the most useful set-theoretic axioms should be those which are also useful for non-set-theoretic mathematics. Finally, he formulated the conjecture that the intrinsically justified higher-order principles of HP will prove useful to find first-order axioms which are useful for non-set-theoretic mathematics: such axioms will then have to be considered true axioms of set theory.

The conference had several outcomes. First, we believe it helped understand some of the underlying assumptions in the HP, and also the theoretical challenges it has to face up to, and the different ways such challenges can be met (Arrigoni, Weydert, Barton).

The potentialist conception was reviewed in depth (Linnebo, Hellman), and its main advantages in light of the clear foundational purposes of set theory fully described.

A description of the multiverse and its utility for set theory was carried out in several talks, and the pressing issues of truth and ontology relating to it, as arising from pluralism (Waxman) or in relation to realism (Venturi), were also examined.

Naturalism was evoked in some talks (Leng, Venturi, Arrigoni). From these talks and the ensuing discussion, it seems reasonable to assert that it is still unclear whether naturalism can properly ground set-theoretic work in a fully satisfactory way, especially if one shifts to a multiversist conception.

New set-theoretic axioms, arising from the need for `maximality’ principles, were the subject of several talks (Barton, Friedman, Blue, Viale), all of which, in our opinion, helped dispel some confusion relating to the notion of maximality, but, at the same time, also clearly highlighted how poorly understood the notion still is. The debate on what form maximality is more acceptable is still open, but it seems that the HP may have good prospects to break new grounds.

Finally, Sanders’ talk hinted at how the issue of what foundational theory is preferable, among those available, might be solved in a way alternative to those usually discussed, that is, by looking into such features as that of `computability’.

In recognition of the joint effort of the organisers and speakers, a proposal for the proceedings of the conference, also including some of the papers discussed at previous SOTFOMs, will be submitted to Synthese.

A selection of slides for the talks is available here: Weydert, Venturi, Arrigoni, Hellman, Linnebo, Friedman, Sanders, Viale, Barton, Waxman and Warren.

We already look forward to SotFoM IV!

]]>The organisers are delighted to announce the programme for the upcoming conference on `The Hyperuniverse Programme’, part of the Symposia on the Foundations of Mathematics series. The Hyperuniverse Programme was launched in 2012, and is currently pursued within a Templeton-funded research project at the Kurt Gödel Research Center in Vienna. It aims to identify and philosophically motivate the adoption of new set-theoretic axioms.The programme intersects several topics in the philosophy of set theory and of mathematics, such as the nature of mathematical (and set-theoretic) truth, the universe/multiverse dichotomy, the alternative conceptions of the set-theoretic multiverse, the conceptual and epistemological status of new axioms and their alternative justificatory frameworks.The aim of SotFoM III and The Hyperuniverse Programme Joint Conference is to bring together scholars who, over the last years, have contributed mathematically and philosophically to the ongoing work and debate on the foundations and the philosophy of set theory, in particular, to the understanding and the elucidation of the aforementioned topics. The three-day conference, taking place September 21-23 at the KGRC in Vienna, will feature invited and contributed speakers.

Programme:

Day 1 – 21 September 2015

1000-1005 Introductory remarks

1005-1135 Tatiana Arrigoni: TBC

1135-1150 Coffee Break

1150-1250 Giorgio Venturi: `Forcing, Multiverse and Realism’

1250-1500 Lunch

1500-1600 Daniel Waxman and Jared Warren: `Is there a good argument for mathematical pluralism?’

1600-1615 Coffee Break

1615-1715 Matteo Viale: `Category forcings and generic absoluteness: Explaining the success of strong forcing axioms.’

1715-1730 Coffee break

1730-1900 Øystein Linnebo: `Potentialism about set theory.’

Day 2 – 22 September 2015

0900-1030 Mary Leng: `On “Defending The Axioms”.’

1030-1045 Coffee Break

1045-1215 Neil Barton: `How the hyperuniverse behaves.’

1230 Trip to Mostalm for social lunch.

Day 3 – 23 September

1000 -1005 Introductory remarks

1005-1135 Geoffrey Hellman: `A height-potentialist multiverse view of set theory.’

1135-1150 Coffee Break

1150-1250 Sam Sanders: `Non-standard analysis as a computational foundation.’

1250-1500 Lunch

1500-1600 Emil Weydert: `A multiverse axiom induction framework.’

1600-1615 Coffee Break

1615-1715 Douglas Blue: `Forcing axioms and maximality as the demand for interpretability.’

1715-1730 Coffee break

1730-1900 Sy-David Friedman: `What are axioms of set theory?’

To register, please send an e-mail to sotfom [at] gmail [dot] com with SOTFOM III REGISTRATION as the subject header.

For more information contact one of:

Carolin Antos: carolin.antos-kuby [at] univie [dot] ac [dot] at

Claudio Ternullo: claudio [dot] ternullo [at] univie [dot] ac [dot] at

John Wigglesworth: jmwigglesworth [at] gmail [dot] com

Neil Barton: barton [dot] n [dot] a [at] gmail [dot] com

Or visit https://sotfom.wordpress.com/

]]>Programme:

Day 1 – 21 September 2015

1000 -1005 Introductory remarks

1005-1135 Tatiana Arrigoni: TBC

1135-1150 Coffee Break

1150-1250 Giorgio Venturi: `Forcing, Multiverse and Realism’

1250-1500 Lunch

1500-1600 Daniel Waxman and Jared Warren: `Is there a good argument for mathematical pluralism?’

1600-1615 Coffee Break

1615-1715 Matteo Viale: `Category forcings and generic absoluteness: Explaining the success of strong forcing axioms.’

1715-1730 Coffee break

1730-1900 Øystein Linnebo: `Potentialism about set theory.’

Day 2 – 22 September 2015

0900-1030 Mary Leng: TBC

1030-1045 Coffee Break

1045-1215 Hugh Woodin: TBC

1230 Trip to Mostalm for social lunch.

Day 3 – 23 September

1000 -1005 Introductory remarks

1005-1135 Geoffrey Hellman: `A Height-Potentialist Multiverse View of Set Theory.’

1135-1150 Coffee Break

1150-1250 Sam Sanders: `Non-standard analysis as a computational foundation.’

1250-1500 Lunch

1500-1600 Emil Weydert: `A multiverse axiom induction framework.’

1600-1615 Coffee Break

1615-1715 Douglas Blue: `Forcing axioms and maximality as the demand for interpretability.’

1715-1730 Coffee break

1730-1900 Peter Koellner: `On the Multiverse Conception of Set.’

To register, please send an e-mail to sotfom [at] gmail [dot] com with SOTFOM III REGISTRATION as the subject header.

For more information contact one of:

Carolin Antos: carolin.antos-kuby [at] univie [dot] ac [dot] at

Claudio Ternullo: claudio [dot] ternullo [at] univie [dot] ac [dot] at

John Wigglesworth: jmwigglesworth [at] gmail [dot] com

Neil Barton: barton [dot] n [dot] a [at] gmail [dot] com

Or visit https://sotfom.wordpress.com/

]]>FINAL CFP: SoTFoM III and The Hyperuniverse Programme, Vienna, September 21-23, 2015.

The Hyperuniverse Programme, launched in 2012, and currently pursued within a Templeton-funded research project at the Kurt Gödel Research Center in Vienna, aims to identify and philosophically motivate the adoption of new set-theoretic axioms.

The programme intersects several topics in the philosophy of set theory and of mathematics, such as the nature of mathematical (set-theoretic) truth, the universe/multiverse dichotomy, the alternative conceptions of the set-theoretic multiverse, the conceptual and epistemological status of new axioms and their alternative justificatory frameworks.

The aim of SotFoM III+The Hyperuniverse Programme Joint Conference is to bring together scholars who, over the last years, have contributed mathematically and philosophically to the ongoing work and debate on the foundations and the philosophy of set theory, in particular, to the understanding and the elucidation of the aforementioned topics. The three-day conference, taking place September 21-23 at the KGRC in Vienna, will feature invited and contributed speakers.

Invited Speakers

T. Arrigoni (Bruno Kessler Foundation)

G. Hellman (Minnesota)

P. Koellner (Harvard)

M. Leng (York)

Ø. Linnebo (Oslo)

W.H. Woodin (Harvard)

+

I. Jané (Barcelona) [TBC]

Call for papers

We invite (especially young) scholars to send their abstracts (of 1’500 words or fewer), addressing one of the following topical strands:

– new set-theoretic axioms

– forms of justification of the axioms and their status within the philosophy of mathematics

– conceptions of the universe of sets

– conceptions of the set-theoretic multiverse

– the role and importance of new axioms for non-set-theoretic mathematics

– the Hyperuniverse Programme and its features

– alternative axiomatisations and their role for the foundations of mathematics

Abstracts should be prepared for blind review and submitted through EasyChair on the following page:

https://easychair.org/conferences/?conf=sotfom3hyp

If there is a paper to back up the abstract (say containing details of proofs, if any) they can be sent to sotfom [at] gmail [dot] com.

We especially encourage female scholars to send us their contributions. Accommodation expenses for contributed speakers will be covered by the KGRC.

Key Dates:

Submission deadline: 15 June 2015 (there will *not* be a deadline extension)

Notification of acceptance: 15 July 2015

For further information, please contact:

sotfom [at] gmail [dot] com

or alternatively one of:

C. Antos

N. Barton

C. Ternullo

J. Wigglesworth

The Hyperuniverse Programme, launched in 2012, and currently pursued within a Templeton-funded research project at the Kurt Gödel Research Center in Vienna, aims to identify and philosophically motivate the adoption of new set-theoretic axioms.

The programme intersects several topics in the philosophy of set theory and of mathematics, such as the nature of mathematical (set-theoretic) truth, the universe/multiverse dichotomy, the alternative conceptions of the set-theoretic multiverse, the conceptual and epistemological status of new axioms and their alternative justificatory frameworks.

The aim of SotFoM III+The Hyperuniverse Programme Joint Conference is to bring together scholars who, over the last years, have contributed mathematically and philosophically to the ongoing work and debate on the foundations and the philosophy of set theory, in particular, to the understanding and the elucidation of the aforementioned topics. The three-day conference, taking place September 21-23 at the KGRC in Vienna, will feature invited and contributed speakers.

Invited Speakers

T. Arrigoni (Bruno Kessler Foundation)

G. Hellman (Minnesota)

P. Koellner (Harvard)

M. Leng (York)

Ø. Linnebo (Oslo)

W.H. Woodin (Harvard)

+

I. Jané (Barcelona) [TBC]

Call for papers

We invite (especially young) scholars to send their papers/abstracts, addressing one of the following topical strands:

– new set-theoretic axioms

– forms of justification of the axioms and their status within the philosophy of mathematics

– conceptions of the universe of sets

– conceptions of the set-theoretic multiverse

– the role and importance of new axioms for non-set-theoretic mathematics

– the Hyperuniverse Programme and its features

– alternative axiomatisations and their role for the foundations of mathematics

Papers should be prepared for blind review and submitted through EasyChair on the following page:

https://easychair.org/conferences/?conf=sotfom3hyp

We especially encourage female scholars to send us their contributions. Accommodation expenses for contributed speakers will be covered by the KGRC.

Key Dates:

Submission deadline: 15 June 2015

Notification of acceptance: 15 July 2015

For further information, please contact:

sotfom [at] gmail [dot] com

or alternatively one of:

C. Antos

N. Barton

C. Ternullo

J. Wigglesworth

Dr. Meadows’s talk led naturally into the second day of the conference, which focused primarily on recent work on set-theoretic foundations. Sy-David Friedman (University of Vienna) discussed his current project, the hyper-universe programme, which aims to find and justify new set-theoretic axioms in order to reduce the incompleteness of ZFC set theory (Zermelo-Fraenkel set theory with the Axiom of Choice). Shivaram Lingamneni (Stanford University/University of California, Berkeley) discussed the possibility of resolving the Continuum Hypothesis, a very natural set-theoretic statement that cannot be resolved by the ZFC axioms. Sam Sanders (University of Ghent / Munich Center for Mathematical Philosophy) gave a presentation arguing that predicativism, a limitative approach to arithmetic, justifies different resources dependent upon the acceptance of infinitesimals (a controversial but useful kind of mathematical object). Finally, the day was rounded off by Victoria Gitman (City University of New York) who explored different kinds of Choice principle (a natural mathematical principle) in the context of class theory, thus informing the debate concerning a philosophically and mathematically fascinating kind of entity linked to the very inception of foundations.

Because of the outstanding quality of the contributed papers, the organisers accepted an additional paper from Benedict Eastaugh (Bristol), who presented the last talk on the morning of the third day. In this presentation, Benedict’s research on reverse mathematics was applied to the problem of discovering new set-existence principles.

On all three days, there was ample time for lively and informal discussion at the organised coffee breaks, lunches and dinners. We were fortunate to have delegates from across the UK, Europe and the US to bring international perspectives on all of the topics discussed in the presentations. We anticipate that these informal discussions will lead to the development of research networks where these issues can be discussed in further detail. Given the high quality of all of the presentations, we plan to arrange for a proceedings volume to collect together a selection of papers based on the talks given at all of the Symposia on the Foundations of Mathematics.

The most important conclusion to be drawn from the conference is that there is still a lot of work to be done in the foundations of mathematics. From the set-theoretic perspective, the search for new axioms, and how to justify them, is well under way. On the category-theoretic/homotopy type theory side, more work is needed to enable philosophers to understand and apply the methods used in these areas. And further work is needed to develop a framework within which we can compare these two approaches (and any others) in the foundations of mathematics.

We hope to tackle these projects in further instalments of the Symposia on the Foundations of Mathematics. Based on the success of this conference in London, there are plans for two more Symposia. The next event will take place in Vienna, at the Kurt Gödel Research Center, in August/September 2015. And there are plans for a subsequent event to take place in Bristol during the Summer 2016. There has been an enthusiastic response to the idea of further events to continue these discussions.

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