Institute for Mathematical Sciences Event Archive
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IMS-JSPS Joint Workshop in Mathematical Logic and Foundations of Mathematics
(1 - 5 September 2014)
Organizing Committee · Visitors and Participants · Overview · Activities · Venue
- Chi Tat Chong (National University of Singapore)
- Frank Stephan (National University of Singapore)
- Kazuyuki Tanaka (Tohoku University)
- Yue Yang (National University of Singapore)
This workshop is jointly sponsored by the Japan Society for the Promotion of Science and the National University of Singapore.  In recent years, the interaction among researchers in East Asia, particularly in Japan, China and Singapore in foundations and other areas of mathematical logic has increased significantly. Two successful workshops in computability theory (recursion theory) and the foundations of mathematics were held in the city of Tokyo in 2013 and 2014. This workshop is intended to provide a venue for continued interaction and to serve as an opportunity to explore new research collaborations in three broad areas of common interest: reverse mathematics (involving both standard and nonstandard models of arithmetic), algorithmic randomness (in both classical and higher setting), and set theory (particularly cardinal characters of the continuum).
The central theme of reverse mathematics is to investigate the proof-theoretic strength of a mathematical theorem. Its history goes back to Hilbert's program on the foundations of mathematics, and techniques in recursion theory have been successfully introduced to investigate problems in reverse mathematics. As examples, there have been major developments in the study of combinatorial principles related to Ramsey's theorem for pairs, resolving several outstanding problems in reverse mathematics. The techniques involved both standard and nonstandard models of arithmetic.
Randomness is a mathematical concept that spans over a broad class of mathematical objects from finite words to transfinite cardinals.
Classical probability theory deals with distributions of randomly chosen objects; however, it does not formulate or even allow for a definition of an individual random object. There is an enduring appeal to the intuitive notion of a real number or of an infinite binary sequence which looks as chosen at random. This intuitive notion can be made more precise when it is interpreted within an effective framework:
no effective test detects any regularity in the real number.
An elegant and useful characterisation of the randomness of an infinite binary sequence R is given by the algorithmic incompressibility of its finite initial segments:
A infinite binary sequence R is random if for almost every finite initial segment I of R there does not exist any description of I (for example, by means of a computer program) which is essentially shorter than the initial segment itself. This approach permits one to calibrate degrees of randomness, characterise applications of randomness and prove preservation of randomness across type, such as between real numbers on the line and sample paths in Brownian motion. In addition, ideas and techniques from effective descriptive set theory may be introduced to investigate randomness from the point of view of second order definability.
The broad area of set theory of the reals has seen a lot of recent activity by researchers in Japan and in Singapore. One topic of interest within this area is cardinal characteristics of the continuum, which capture certain combinatorial features of the set of real numbers. There have a number of recent advances in this field, most notably the proof by Malliaris and Shelah that p = t. Closely related to this is the study of the descriptive complexity of certain maximal families of sets of reals, such as almost disjoint families and ultrafilters. Recent work on this has been done by both sides. Another broad area of mutual interest is the study of limitations of forcing axioms such as PFA and MM and their consequences such as the P-ideal dichotomy.
08:50am - 09:10am |
Registration |
09:10am - 09:15am |
Opening Remarks Chi Tat Chong, National University of Singapore |
09:15am - 10:00am |
Variants of infinite games and their strength |
10:00am - 10:30am |
--- Group Photo & Coffee Break --- |
10:30am - 11:15am |
The theory of universally Baire sets in 2^{\omega_1} (PDF) |
11:15am - 12:00pm |
Some fixed point theorems and reverse mathematics |
12.00pm - 01:30pm |
--- Lunch --- |
01:30pm - 02:15pm |
Some remarks on computation on reals (PDF) |
Tuesday, 2 Sep 2014 |
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09:00am - 09:15am |
Registration |
09:15am - 10:00am |
A new kind of computability theory on reals (PDF) |
10:00am - 10:30am |
--- Coffee Break --- |
10:30am - 11:15am |
Coloring on trees and Ramsey's theorem for pairs (PDF) |
11:15am - 12:00pm |
Measuring the relative strength of sets of natural numbers (PDF) |
12.00pm - 01:30pm |
--- Lunch Reception at IMS --- |
01:30pm - 02:15pm |
Lindelöf group with non-Lindelöf square and strong negative partition relation (PDF) |
02:15pm - 03:00pm |
Non-principal ultrafilters, program extraction and higher order reverse mathematics (PDF) |
Wednesday, 3 Sep 2014 |
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09:00am - 09:15am |
Registration |
09:15am - 10:00am |
Definability of the ground model and large cardinals (PDF) |
10:00am - 10:30am |
--- Coffee Break --- |
10:30am - 11:15am |
Termination theorem and Ramsey's theorem (PDF) |
11:15am - 12:00pm |
Beyond Peano arithmetic? (PDF) |
12.00pm - 01:30pm |
--- Lunch --- |
Thursday, 4 Sep 2014 |
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09:00am - 09:15am |
Registration |
09:15am - 10:00am |
Some properties of probability theory in reverse mathematics (PDF) |
10:00am - 10:30am |
--- Coffee Break --- |
10:30am - 11:15am |
Randomness in the absence of full induction |
11:15am - 12:00pm |
Whitehead problem and ACA_0 (PDF) |
12.00pm - 01:30pm |
--- Lunch --- |
01:30pm - 02:15pm |
Intuitionistic provability and uniformly provability in RCA (PDF) |
02:15pm - 03:00pm |
Some RF-type theorems in reverse mathematics (PDF) |
Friday, 5 Sep 2014 |
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09:00am - 09:15am |
Registration |
09:15am - 10:00am |
Weak lowness notions for Kolmogrov complexity (PDF) |
10:00am - 10:30am |
--- Coffee Break --- |
10:30am - 11:15am |
Expressibility of simple unary generalized quantifier (PDF) |
11:15am - 12:00pm |
On exact d.c.e. degrees |
12.00pm - 01:30pm |
--- Lunch --- |
01:30pm - 02:15pm |
On embedding certain partial orders into the P-points under Tukey and RK reducibility (PDF) |
02:15pm - 03:00pm |
The role of the axiom DOM in reverse mathematics and its applications to model inductive inference |
Students and researchers who are interested in attending these activities are requested to complete the online registration form.
The following do not need to register:
- Those invited to participate.
Organizing Committee · Visitors and Participants · Overview · Activities · Venue
