This page is three and a half years out-of-date.

To everyone who follows my research - I apologise! I will try to update this webpage (or build an entirely new page) ASAP, around Easter 2010. New papers, new preprints, totally new explanations and "programmatic" texts will be added. If you want to see my recent preprints - just send me an email.

 

My research motivation and my explanations have evolved very much, so the new page will be completely different. I'll add all the thinigs that accummulated in the last three years: unprovability and logical strength in number theory, some new Boolean Relation Theory theorems (the most successful of all H.Friedman's templates so far), a programmatic article on Arithmetical Splitting (+templates), an article on zeta with Andreas Weiermann, three new preprints with Michiel De Smet (polynomials, Nash-Williams-theoretic indiscernibles/unprovability and Impredicative Ramsey Theory), a short pamphlet ``The Platonism Delusion" and my Bristol Logic Notes.

I'll aslo write some explanations about the three other ongoing projects:

below is old material (a lot is out-of date) from three years ago.

Unprovability and logical strength in mathematics

Finished pieces from the past:
Exact unprovability results for compound combinatorial classes. (2009). Annals of Pure and Applied Logic, 157 , pp. 77-84.
Unprovable statements based on diophantine approximation and distribution of values of zeta-functions. (2007). Joint with Andreas Weiermann. Preprint. (this is a short version)
Long sequences of braids. (2007). With Lorenzo Carlucci. Will not be published. Was subsumed by a later paper by Carlucii, Dehornoy and Weiermann.
Unprovability of sharp versions of Friedman's sine-principle. (2007). Proceedings of the American Mathematical Society, 135, pp. 2967 - 2973. available online.
The strength of infinitary ramseyan principles can be accessed by their densities (with Andreas Weiermann) (2005). Accepted for publication in Annals of Pure and Applied Logic.
Brief introduction to unprovability. (2009). Logic Colloquium 2006, Lecture Notes in Logic, pp. 38 -64.
Several proofs of PA-unprovability. (2005). Contemporary Mathematics Series of the AMS, 380, pp. 29-43.

Model theory and models of arithmetic

I originally studied Model Theory (models of arithmetic) as the main subject of my PhD research. My PhD results (on order-types of models of arithmetic) are unpublished but you can download the file below. My main achievments in this subject are: a large body of results about the linear order structure of models of Peano Arithmetic, including nonclassifiability results (non-structure arguments), ultimate results on order-types of models of PA strongly interpreted in other models of PA, partial results on the famous problem of whether order-types of all nonstandard models of PA are the same for all completions of PA, a study of arithmetically saturated models, and, recently, a study of first-order theories, whose models are treatable or categorically treatable (intuitively: resembling unique countable models (of omega-categorical theories) or unique countable recursively saturated models) and my recent article below that uses internalized constructions to give sufficient and necessary Sigma_1^1 conditions for a structure to be resplendent or chronically resplendent. The main idea here is to prove arithmetized versions of constructions of resplendent and recursively saturated models (versions of the arithmetized completeness theorem) and then infer results about models "in the real world".

Resplendent models and Sigma_1^1 definability with an oracle. (2008). Archive for Mathematical Logic, 47, pp. 604 - 621. , available online at Springer
Resplendent linear orders. (with Richard Kaye) (2002). Unpublished Article.
On order-types of models of arithmetic. (with R.Kaye) (2001). Contemporary Mathematics Series of the AMS, 302, pp. 275-285.
Order-types of models of arithmetic and a connection with arithmetic saturation. (2004). Lobachevskii Journal of Mathematics, 16, pp. 3-15. available online
On order-types of models of arithmetic. PhD Thesis. University of Birmingham. (2000).

Drafts.

Here are some drafts and sketches (usually out-of-date). This section is intended for colleagues who are closely following my research. This dungeon is constantly changing and its inhabitants eventually migrate upstairs.
Model Theory of Kruskal's Theorem (2006). a draft is coming at some point. (The result is that if a semi-regular initial segment satisfies the labeled Kruskal Theorem ("if X is wqo then the X-labeled trees are wqo") then this initial segment is strong. An obvious (known) corollary is that labeled Kruskal implies all of ACA_0.)
Long games on braids. Joint with Lorenzo Carlucci. (2006).
The strength of infinitary principles can be accessed by their densities (with Andreas Weiermann) (2006). (this is an old out-of-date draft, but a new draft will be coming some time in 2008)
Logical indeterminacy in the quality theory of dynamical systems (with Sergey Kryzhevich). We are building upon my article about the sine-principle (see above). There are some raw preliminary results.
Finite Canonical Ramsey Theorem gives rise to four unprovable statements. this is a not very hard application of the theory of indicators, will be typed eventually.
Exact unprovability thresholds for graph minors. , a first rough draft about unprovability threshold for the graph minor theorem.

Non-papers:

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