Reverse Mathematics: Constructivism and Combinatorics
Research supported by the John Templeton Foundation
The research described below will be supported in part or entirely
by Grant ID# 20800 from the John Templeton Foundation during the
period from June 1, 2011 to July 31, 2012. Any opinions expressed
here or in the linked materials are those of the author and do not
necessarily reflect the views of the John Templeton Foundation.
Executive Summary
From the project proposal:
This project will contribute to the program of reverse mathematics
by proving new reverse mathematics results in the area of combinatorics
and by extending our understanding of the applicability of reverse mathematics
to questions in constructive mathematics. The work addresses foundational
questions in the mathematical sciences, specifically,
“
What are the limits of mathematics in advancing human knowledge?
”
Results will contribute to our understanding of the limits of
mathematics within mathematics. By delimiting the nature and
capabilities of formal reasoning, we can better understand and
appreciate the potential of other, less formal means of gaining knowledge.
Funding will free Hirst from teaching duties and support some travel for
dissemination. Foundation support paired with sabbatical support from
Appalachian State and donated summer research will provide a 14 month
period devoted entirely to this research. Expected outcomes include at
least two research papers on the central projects, additional related
research papers, and various presentations.
Two main projects are proposed. The first is extension of uniformization
results that extract implications for constructive mathematics from
results in reverse mathematics. The second is formalization of
ultrafilter proofs of Hindman's theorem.
Presentations

A pdf file of the
slides
for
Three tales about Weak Konig's Lemma,
a talk given in the logic seminar
at the Ohio State University on November 20, 2012.

A pdf file of the
slides
for
Reverse mathematics and field extensions
a talk given at the Association for Symbolic Logic
2012 North American Annual Meeting on
April 1, 2012.

A pdf file of the
slides
for
Reverse mathematics and dichotomy
a talk given at the Joint Mathematics Meetings in Boston on
January 6, 2012.

A pdf file of the
slides
for
Reverse mathematics and persistent reals
a talk given at the Midwest Computability Seminar X on November 1, 2011.

A pdf file of the
slides
for
Two familiar principles in disguise
a talk given in the Notre Dame Logic Seminar on October 27, 2011.

A pdf file of the slides for
Two combinatorial proofs and some related questions
a talk given at the Reverse Mathematics Workshop
at the University of Chicago on September 17, 2011.

A pdf file of the slides for
Reverse Mathematics: Constructivism and Combinatorics,
a talk given at Foundational Questions in the Mathematical Sciences,
a meeting sponsored by the John Templeton Foundation at the
International Academy Traunkirchen, Austria on July 812, 2011.
Publications

On uniform relationships between combinatorial problems,
with François Dorais, Damir Dzhafarov, Joseph Mileti, and Paul Shafer,
Transactions of the American Mathematical Society,
368 (2016) 13211359.
Available online
(DOI) 10.1090/tran/6465 for subscribers.
A draft is available.

Comparing the strength of diagonally nonrecursive functions
in the absence of Σ^{0}_{2} induction,
with François Dorais and Paul Shafer, The Journal of Symbolic Logic,
80 no. 4, (2015) 12111235.
Available online
http://journals.cambridge.org/abstract_S0022481215000432
for subscribers.
A draft is available.

Generics for computable Mathias forcing,
with Peter Cholak, Damir Dzhafarov, and Theodore Slaman, Annals of
Pure and Applied Logic 165:9, (2014) 14181428.
Available online
(DOI) 10.1016/j.apal.2014.04.011.
A draft is available.

Reverse mathematics and algebraic field extensions,
with François Dorais and Paul Shafer, Computability 2:2, (2013) 7592.
Available online
(10.3233/COM13021).
A draft is available.

Disguising induction: Proofs of the pigeonhole principle for trees,
in: Foundational Adventures: Essays
in Honor of Harvey M. Friedman, (Neil Tennant, editor)
Templeton Press (2012).
Available online at
foundationaladventures.wordpress.com.
A draft is available in pdf format.

Reverse mathematics, trichotomy, and dichotomy,
with François Dorais and Paul Shafer, Journal of Logic and Analysis
4:13, (2012) 114.
Available online
(DOI: 10.4115/jla.2012.4.13).
A draft is available in pdf format.

On Mathias generic sets,
with Peter Cholak and Damir Dzhafarov, pages 129138 in:
How the World Computes: Proceedings of the Turing Centenary
Conference and 8th Conference on Computability in Europe, CiE 2012,
LNCS 7318, (Cooper, Dawar, and Lowe, editors)
SpringerVerlag (2012) ISBN: 9783642308697.
Available online
(DOI: 10.1007/9783642308703_14).
A draft is available in pdf format.

More reverse mathematics of the HeineBorel theorem,
with Jessica Miller, Journal of Logic and Analysis
4:6, (2012) 110.
Available online
(DOI: 10.4115/jla.2012.4.6).
A draft is available in pdf format.

Hilbert versus Hindman,
Archive for Mathematical Logic
51:12, (2012) 123125.
Available online
(DOI: 10.1007/s0015301102574) for subscribers.
A draft is available in pdf format.
Other Activities and Materials
 The
web page
for the AMS/ASL joint special session on the Life and Legacy of Alan Turing,
to be held in January at the Joint Mathematics Meetings in Boston.
 A
press release
from the University News service at Appalachian State University.