18 research outputs found
Iterative forcing and hyperimmunity in reverse mathematics
The separation between two theorems in reverse mathematics is usually done by
constructing a Turing ideal satisfying a theorem P and avoiding the solutions
to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a
forcing technique for iterating a computable non-reducibility in order to
separate theorems over omega-models. In this paper, we present a modularized
version of their framework in terms of preservation of hyperimmunity and show
that it is powerful enough to obtain the same separations results as Wang did
with his notion of preservation of definitions.Comment: 15 page
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
Unprovability and phase transitions in Ramsey theory
The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions.
Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas.
Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions.
Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem.
Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability.
Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families.
Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details
The proof-theoretic strength of Ramsey's theorem for pairs and two colors
Ramsey's theorem for -tuples and -colors () asserts
that every k-coloring of admits an infinite monochromatic
subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and
two colors, namely, the set of its consequences, and show that
is conservative over . This
strengthens the proof of Chong, Slaman and Yang that does not
imply , and shows that is
finitistically reducible, in the sense of Simpson's partial realization of
Hilbert's Program. Moreover, we develop general tools to simplify the proofs of
-conservation theorems.Comment: 32 page
The weakness of being cohesive, thin or free in reverse mathematics
Informally, a mathematical statement is robust if its strength is left
unchanged under variations of the statement. In this paper, we investigate the
lack of robustness of Ramsey's theorem and its consequence under the frameworks
of reverse mathematics and computable reducibility. To this end, we study the
degrees of unsolvability of cohesive sets for different uniformly computable
sequence of sets and identify different layers of unsolvability. This analysis
enables us to answer some questions of Wang about how typical sets help
computing cohesive sets.
We also study the impact of the number of colors in the computable
reducibility between coloring statements. In particular, we strengthen the
proof by Dzhafarov that cohesiveness does not strongly reduce to stable
Ramsey's theorem for pairs, revealing the combinatorial nature of this
non-reducibility and prove that whenever is greater than , stable
Ramsey's theorem for -tuples and colors is not computably reducible to
Ramsey's theorem for -tuples and colors. In this sense, Ramsey's
theorem is not robust with respect to his number of colors over computable
reducibility. Finally, we separate the thin set and free set theorem from
Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of
thin set theorems in reverse mathematics. This shows that in reverse
mathematics, the strength of Ramsey's theorem is very sensitive to the number
of colors in the output set. In particular, it enables us to answer several
related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page
Phase Transitions for Gödel Incompleteness
Gödel's first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers
have been looking for natural examples of such assertions and breakthroughs have been obtained in the seventies by Jeff Paris (in part jointly with Leo Harrington and Laurie Kirby) and Harvey Friedman who produced first mathematically interesting
independence results in Ramsey theory (Paris) and well-order and well-quasi-order theory (Friedman).
In this article we investigate Friedman style principles of combinatorial well-foundedness for the ordinals below epsilon_0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.
For these independence principles we classify (as a part of a general research program) their phase transitions, i.e. we classify exactly the bounding conditions which lead from
provability to unprovability in the induced combinatorial
well-foundedness principles.
As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel's coding of finite sequences from his classical 1931 paper on his incompleteness results.
This choice makes the investigation highly non trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions.
For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem by Parameswaran which apparently is far remote from Gödel's theorem