STRONG REDUCTIONS BETWEEN COMBINATORIAL PRINCIPLES

2016 ◽  
Vol 81 (4) ◽  
pp. 1405-1431 ◽  
Author(s):  
DAMIR D. DZHAFAROV
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
New Techniques ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Problems And Solutions ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Image Position

AbstractThis paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$, and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$, or strongly computably reducible to $SRT_2^2$. The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

2014 ◽  
Vol 20 (2) ◽  
pp. 170-200 ◽  
Author(s):  
C. T. CHONG ◽  
WEI LI ◽  
YUE YANG
Keyword(s):  
Recursion Theory ◽  
Reverse Mathematics ◽  
Second Order ◽  
Ramsey's Theorem ◽  
Second Order Arithmetic ◽  
Ramsey’S Theorem ◽  
Nonstandard Models ◽  
Combinatorial Principles ◽  
Order Arithmetic ◽  
Effective Computability

AbstractWe give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

scholarly journals THE DEFINABILITY STRENGTH OF COMBINATORIAL PRINCIPLES

2016 ◽  
Vol 81 (4) ◽  
pp. 1531-1554 ◽  
Author(s):  
WEI WANG
Keyword(s):  
Combinatorial Problem ◽  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
Combinatorial Principle ◽  
Definable Set ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Definition Of ◽  
Image Position

AbstractWe introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey’s Theorem for colorings of pairs could help in simplifying the definitions of some ${\rm{\Delta }}_2^0$ sets, while some others could not. We also investigate some consequences of Ramsey’s Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.

scholarly journals RANDOMNESS NOTIONS AND REVERSE MATHEMATICS

2019 ◽  
Vol 85 (1) ◽  
pp. 271-299
Author(s):  
ANDRÉ NIES ◽  
PAUL SHAFER
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Existence Principle ◽  
Order Arithmetic ◽  
Image Position

AbstractWe investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal.

scholarly journals Reverse mathematics and the equivalence of definitions for well and better quasi-orders

2004 ◽  
Vol 69 (3) ◽  
pp. 683-712 ◽  
Author(s):  
Peter Cholak ◽  
Alberto Marcone ◽  
Reed Solomon
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
Base System ◽  
Mathematical Concepts ◽  
Second Order Arithmetic ◽  
Finite Set ◽  
Combinatorial Principles ◽  
Quasi Order ◽  
Order Arithmetic ◽  
Better Quasi Order

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Richard A. Shore
Keyword(s):  
Reverse Mathematics ◽  
The Other ◽  
Base System ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Points Of View ◽  
The One ◽  
Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

Reverse Mathematics: The Playground of Logic

2010 ◽  
Vol 16 (3) ◽  
pp. 378-402 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Reverse Mathematics ◽  
Computational Approach ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Base Theory ◽  
Order Arithmetic

AbstractThis paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

Extension of Lifschitz' realizability to higher order arithmetic, and a solution to a problem of F. Richman

1991 ◽  
Vol 56 (3) ◽  
pp. 964-973 ◽  
Author(s):  
Jaap van Oosten
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Church’S Thesis ◽  
Second Order Arithmetic ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Image Position

AbstractF. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic HAH:and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of HAH in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP.In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that RP is derivable in HAH + CT + MP + ECT0, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

scholarly journals HYPERCLASS FORCING IN MORSE-KELLEY CLASS THEORY

2017 ◽  
Vol 82 (2) ◽  
pp. 549-575 ◽  
Author(s):  
CAROLIN ANTOS ◽  
SY-DAVID FRIEDMAN
Keyword(s):  
Second Order ◽  
Inaccessible Cardinal ◽  
Second Order Arithmetic ◽  
Class Theory ◽  
Order Arithmetic ◽  
Image Position ◽  
Class Forcing

AbstractIn this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK**. We define this forcing by using a symmetry between MK** models and models of ZFC− plus there exists a strongly inaccessible cardinal (called SetMK**). We develop a coding between β-models ${\cal M}$ of MK** and transitive models M+ of SetMK** which will allow us to go from ${\cal M}$ to M+ and vice versa. So instead of forcing with a hyperclass in MK** we can force over the corresponding SetMK** model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK** model M+ can be forced to look like LK*[X], where κ* is the height of M+, κ strongly inaccessible in M+ and $X \subseteq \kappa$. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK**, which will in turn be an extension of the original ${\cal M}$. Our main result combines hyperclass forcing with coding methods of [3] and [4] to show that every β-model of MK** can be extended to a minimal such model of MK** with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.

scholarly journals INTERPRETATIONS BETWEENω-LOGIC AND SECOND-ORDER ARITHMETIC

2014 ◽  
Vol 79 (3) ◽  
pp. 845-858
Author(s):  
RICHARD KAYE
Keyword(s):  
Model Theory ◽  
Second Order ◽  
Standard System ◽  
Technical Result ◽  
Second Order Arithmetic ◽  
Nonstandard Model ◽  
Order Arithmetic ◽  
Single Formula ◽  
Image Position

AbstractThis paper addresses the structures (M, ω) and (ω, SSy(M)), whereMis a nonstandard model of PA andωis the standard cut. It is known that (ω, SSy(M)) is interpretable in (M, ω). Our main technical result is that there is an reverse interpretation of (M, ω) in (ω, SSy(M)) which is ‘local’ in the sense of Visser [11]. We also relate the model theory of (M, ω) to the study of transplendent models of PA [2].This yields a number of model theoretic results concerning theω-models (M, ω) and their standard systems SSy(M, ω), including the following.•$\left( {M,\omega } \right) \prec \left( {K,\omega } \right)$if and only if$M \prec K$and$\left( {\omega ,{\rm{SSy}}\left( M \right)} \right) \prec \left( {\omega ,{\rm{SSy}}\left( K \right)} \right)$.•$\left( {\omega ,{\rm{SSy}}\left( M \right)} \right) \prec \left( {\omega ,{\cal P}\left( \omega \right)} \right)$if and only if$\left( {M,\omega } \right) \prec \left( {{M^{\rm{*}}},\omega } \right)$for someω-saturatedM*.•$M{ \prec _{\rm{e}}}K$implies SSy(M, ω) = SSy(K, ω), but cofinal extensions do not necessarily preserve standard system in this sense.• SSy(M, ω)=SSy(M) if and only if (ω, SSy(M)) satisfies the full comprehension scheme.• If SSy(M, ω) is uniformly defined by a single formula (analogous to aβfunction), then (ω, SSy(M, ω)) satisfies the full comprehension scheme; and there are modelsMfor which SSy(M, ω) is not uniformly defined in this sense.

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