Topic Archive: reverse mathematics
The unreasonable effectiveness of Nonstandard Analysis.
Nonstandard Analysis (NSA) was introduced around 1965 by Robinson as a formalization of the intuitive infinitesimal calculus which is in use to date in most of physics and historically in mathematics until the advent of Weierstrass’ epsilon-delta framework. Famous people like Connes and Bishop have derided NSA for its alleged utter lack of computational/effective/constructive content. In this talk I show that every theorem of ‘pure’ NSA can be (equivalently) converted to a theorem of computable mathematics. In many cases, the resulting theorem is even constructive in the sense of Bishop.
Weihrauch reducibility and Ramsey theorems
Weihrauch reducibility is a common tool in computable analysis for understanding and comparing the computational content of theorems. In recent years, variations of Weihrauch reducibility have been used to study Ramsey type theorems in the context of reverse mathematics where they give a finer analysis than implications in RCA0 and they allow comparisons of computably true principles. In this talk, we will give examples of recent results and techniques in this area.
NERDS on April 2, 2016
The Spring 2016 New England Recursion and Definability Seminar (“NERDS”) will take place on Saturday, April 2, 2016 at Springfield College, in Springfield, MA. Further details and abstracts of talks will be posted on nylogic.org as they become available.
Computing uniform (metastable) rates of convergence from the statement of the theorem alone
Consider a convergence theorem of the following form.
(T): If the property P holds, then the sequence (xn) converges.
For example, the monotone convergence principle states that any monotone, bounded sequence of reals converges.
This talk concerns the notion of metastable rates of convergence. The advantage of this notion is that metastable rates are often uniform and computable. Kohlenbach, using proof theory, showed that if the property P is of a certain form and the theorem (T) is provable in a certain formal type theory, then the rate of metastable convergence is both uniform and computable. Avigad and Iovino, using model theory, showed that if the theorem (T) is true and the property P is preserved by ultraproducts, then the rate of metastable convergence is uniform (no mention of computability). In this result, using computable analysis and computable continuous model theory, we show that if (T) is true and P is axiomatizable in continuous logic, then the corresponding metastable bounds are both uniform and computable from P. This generalizes both of the previous results.
Ramsey’s Theorem applied to infinite traceable graphs
We consider three applications of Ramsey’s Theorem to infinite traceable graphs and finitely generated lattices from the point of view of reverse mathematics. For two of the applications, we will show that Ramsey’s Theorem is necessary while for the third application, it is not necessary. We will conclude with some related open questions.
The Autumn 2014 meeting of NERDS, the New England Recursion & Definability Seminar, will take place on Saturday, October 18 at Assumption College, in Worcester, MA, beginning at 10:30 a.m. The principal organizers are Brooke Andersen, Damir Dzhafarov, and Reed Solomon. All talks are posted on nylogic.org (find NERDS under the “Conferences” tab), and they will take place in the Carriage House building on the campus of Assumption College. Directions.
Computable algebra: a personal perspective
I will give a brief overview of some of my recent research in the field of Computable Algebra, emphasizing connections between Computability Theory and Algebra. Some of the topics that I hope to cover include Artinian and Euclidean rings, as well as infinite dimensional vector spaces. There will be no proofs, just statements of theorems along with discussions on their logical and algebraic significance.
New directions in reverse mathematics
Mathematics today benefits from having “firm foundations,” by which we usually mean a system of axioms sufficient to prove the theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were asking it about Euclid’s geometry. Reverse mathematics provides a modern approach to this kind of question. A striking fact repeatedly demonstrated in this area is that the vast majority of mathematical propositions can be classified into just five main types, according to which set-existence axioms are needed to carry out their proofs. But more recently, a growing number of principles falling outside this classification have emerged, whose logical strength is more difficult to understand. These turn out to include many important mathematical results, such as various combinatorial problems related to Ramsey’s theorem, and several set-theoretic equivalents of the axiom of choice. I will discuss some of these “irregular” principles, and some new approaches that have arisen from trying to understand why their strength is so different from that of most other theorems. In particular, this investigation reveals new connections between different mathematical areas, and exposes the rich and complex combinatorial and algorithmic structure underlying mathematics as a whole.
Higher-order Reverse Mathematics: Where existence meets computation via infinitesimals.
Classically, the existence of an object tells us very little about how to construct said object.
We consider a nonstandard version of Ulrich Kohlenbach’s higher-order Reverse Mathematics
in which there is a very elegant and direct correspondence between, on one hand, the existence
of a functional computing an object and, on the other hand, the classical existence of this object
with the same standard and nonstandard properties. We discuss how these results -potentially-
contribute to the programs of finitistic and predicativist mathematics.
Models of Reverse Mathematics
We discuss two results relating ideas in Reverse Mathematics to the properties of models of first order arithmetic. The first shows that we can extend second order arithmetic by the existence of a non-principal ultrafilter — a third order property — while remaining conservative. The second result shows that we can extend models of RCA so that any particular set is definable; this allows us to recover some properties of models of Peano arithmetic for models of RCA.