Completeness is overrated
It is a common theme in constructive mathematics that adding completeness to the assumptions often allows us to prove otherwise constructively unprovable statements. In this talk we show that we actually only need a very weak version of completeness is enough. We also show that there is an enormous amount of spaces that aren’t complete, but satisfy the weakened version of completeness, thus generalising many theorems of constructive analysis. Link to the slides.
Computing with Infinite Objects
Konstruktivismus–Geschichte, Philosophie und Praxis
Variations on a theme by Ishihara:
Variations of this talk have been presented at the University of Krakow (Dec 2010), the University of Bonn (2011) and more refined at the University of Cape Town, UNISA in Pretoria (2012). A preliminary version of a paper can be found here.
Information on Andrej Bauer's infinite time Turing machine model in which there is an injection from Baire space to the natural numbers are on his blog. An overview of constructive Zermelo Fraenkel set theory (CZF) here.
Principles weaker than BD-N:
Oberwolfach Nov. 2011. A five minutes talk on work in progress. Slides for this talk here.
Constructions and Constructive Reverse Mathematics:
Slides for this talk here.
A constructive look at the Vitali Covering Theorem
At the wonderful "Workshop on Constructive Aspects of Logic and Mathematics" in Kanazawa, Japan in March 2010 I presented work on constructive measure theory. This is work in progress with Anton Hedin from Uppsala University, Sweden.
The Fan Theorem(s):
In this talk, versions of which were given at the CCC09 meeting in Cologne and the 25th Summer Topology Conference in Kielce, I talked about my favourite topic: Brouwer's fan theorems. The slides of the newer version can be found here.
The dark/light side of constructive reverse mathematics:
The "dark side" was presented at the 2007 NZMASP in Queenstown and won the NZIMA Best Presentation Award. Having done so well the talk was recycled for the 2007 joint meeting of NZMS and AMS in Wellington.
For the public seminar traditionally given on the same day of the PhD defense at Canterbury University, I presented the prequel talk "The light side of constructive reverse mathematics".
The platypus and the mathematician:
This talk was first given at the 2006 SIMASP in Queenstown. A much more polished version was given at the 2009 joint meeting of the Australian and New Zealand Math Societies. It includes no serious research, but frontal lobotomies, black holes and answered the question to what the correct plural of platypus really is.
Mathematicians generally like to insist that their field is not science, since it is deductive rather than inductive. In this talk we argue that, nevertheless, there is a healthy dose of inductive reasoning in the way mathematicians think.
This talk was given at the 2006 conference Trends in Constructive Mathematics
The results presented have since been published under the same title ‘generalising compactness’.
The constructive content of Sarkovski's theorem:
This talk was given at the 2006 NZMS meeting in Hamilton
The talk contained a short introduction to Sarkovski's theorem, which is everything one can possible hope for in a theorem: easy to state, unexpected, and it comes with a clever proof. Unfortunately there are some concerns about the constructiveness — we can show that it is equivalent to the essentially non-constructive principle LLPO. Nevertheless, as usual, we can fix this shortcoming with minor tweaks. The results presented are currently awaiting urgent attention.