scholarly journals TRANSFINITE RECURSION IN HIGHER REVERSE MATHEMATICS

2015 ◽  
Vol 80 (3) ◽  
pp. 940-969 ◽  
Author(s):  
NOAH SCHWEBER
Keyword(s):  
Reverse Mathematics ◽  
Higher Order ◽  
Steel Type ◽  
Base Theory ◽  
Higher Types ◽  
Flexible Framework ◽  
Image Position ◽  
Transfinite Recursion

AbstractIn this paper we investigate the reverse mathematics of higher-order analogues of the theory $$ATR_0$$ within the framework of higher order reverse mathematics developed by Kohlenbach [11]. We define a theory $$RCA_0^3$$, a close higher-type analogue of the classical base theory $$RCA_0$$ which is essentially a conservative subtheory of Kohlenbach’s base theory $$RCA_{\rm{0}}^\omega$$. Working over $$RCA_0^3$$, we study higher-type analogues of statements classically equivalent to $$ATR_0$$, including open and clopen determinacy, and examine the extent to which $$ATR_0$$ remains robust at higher types. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel’s tagged tree forcing; in the presentation of this result, we develop a new, more flexible framework for Steel-type forcing.

scholarly journals Foundational and Mathematical Uses of Higher Types

1999 ◽  
Vol 6 (31) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Reverse Mathematics ◽  
Continuous Functions ◽  
Higher Order ◽  
Second Order ◽  
Polish Spaces ◽  
Primitive Recursive Arithmetic ◽  
Higher Types ◽  
A Priority ◽  
Primitive Recursive

In this paper we develop mathematically strong systems of analysis in<br />higher types which, nevertheless, are proof-theoretically weak, i.e. conservative<br />over elementary resp. primitive recursive arithmetic. These systems<br />are based on non-collapsing hierarchies (Phi_n-WKL+, Psi_n-WKL+) of principles<br />which generalize (and for n = 0 coincide with) the so-called `weak' K¨onig's<br />lemma WKL (which has been studied extensively in the context of second order<br />arithmetic) to logically more complex tree predicates. Whereas the second<br />order context used in the program of reverse mathematics requires an encoding<br />of higher analytical concepts like continuous functions F : X -> Y between<br />Polish spaces X, Y , the more flexible language of our systems allows to treat<br />such objects directly. This is of relevance as the encoding of F used in reverse<br />mathematics tacitly yields a constructively enriched notion of continuous functions<br />which e.g. for F : N^N -> N can be seen (in our higher order context) to be equivalent<br /> to the existence of a continuous modulus of pointwise continuity.<br />For the direct representation of F the existence of such a modulus is<br />independent even of full arithmetic in all finite types E-PA^omega plus quantifier-free<br />choice, as we show using a priority construction due to L. Harrington.<br />The usual WKL-based proofs of properties of F given in reverse mathematics<br />make use of the enrichment provided by codes of F, and WKL does not seem<br />to be sufficient to obtain similar results for the direct representation of F in<br />our setting. However, it turns out that   Psi_1-WKL+ is sufficient.<br />Our conservation results for (Phi_n-WKL+,  Psi_n-WKL+) are proved via a new<br />elimination result for a strong non-standard principle of uniform Sigma^0_1-<br />boundedness<br />which we introduced in 1996 and which implies the WKL-extensions studied<br />in this paper.

scholarly journals ON IDEMPOTENT ULTRAFILTERS IN HIGHER-ORDER REVERSE MATHEMATICS

2015 ◽  
Vol 80 (1) ◽  
pp. 179-193 ◽  
Author(s):  
ALEXANDER P. KREUZER
Keyword(s):  
Reverse Mathematics ◽  
Higher Order ◽  
Hindman’S Theorem ◽  
Image Position ◽  
Idempotent Ultrafilter ◽  
Order Extension

AbstractWe analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics.Let $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ be the statement that an idempotent ultrafilter on ℕ exists. We show that over $ACA_0^\omega$, the higher-order extension of ACA0, the statement $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ implies the iterated Hindman’s theorem (IHT) and we show that $ACA_0^\omega + \left( {{{\cal U}_{{\rm{idem}}}}} \right)$ is ${\rm{\Pi }}_2^1$-conservative over $ACA_0^\omega + IHT$ and thus over $ACA_0^ +$.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

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.

From Hypothesis Testing to Estimating Functionals

2019 ◽  
pp. 185-259
Author(s):  
John Stillwell
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Hypothesis Testing ◽  
Set Theory ◽  
Reverse Mathematics ◽  
Pythagorean Theorem ◽  
Base Theory ◽  
The Axiom Of Choice ◽  
The Right ◽  
Main Ideas

This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.

Irreducibility of Bernoulli Polynomials of Higher Order

1962 ◽  
Vol 14 ◽  
pp. 565-567 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Positive Integer ◽  
Bernoulli Number ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Rational Numbers ◽  
Higher Order ◽  
Image Position

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

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.

Free discontinuity problems generated by singular perturbation: the n−dimensional case

2000 ◽  
Vol 130 (3) ◽  
pp. 449-469 ◽  
Author(s):  
R. Alicandro ◽  
M. S. Gelli
Keyword(s):  
Singular Perturbation ◽  
Hessian Matrix ◽  
Higher Order ◽  
Dimensional Case ◽  
Free Discontinuity Problems ◽  
Limiting Behaviour ◽  
Image Position

We provide an approximation of some free discontinuity problems by local functionals with a singular perturbation of higher order. More precisely, we study the limiting behaviour of energies of the form where Hu denotes the Hessian matrix of u.

scholarly journals THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

2018 ◽  
Vol 83 (2) ◽  
pp. 817-828 ◽  
Author(s):  
ERIC P. ASTOR
Keyword(s):  
Reverse Mathematics ◽  
Previous Article ◽  
Asymptotic Density ◽  
Turing Degrees ◽  
Image Position ◽  
Immune Set

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

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