scholarly journals Higher Order Reverse Mathematics

2000 ◽  
Vol 7 (49) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Finite Type ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Higher Order ◽  
Second Order ◽  
Base System ◽  
Frame Work

In this paper we argue for an extension of the second order frame-work currently used in the program of reverse mathematics to finite types. In particular we propose a conservative finite type extension RCA^omega_0 of the second order base system RCA_0. By this conservation nothing is lost for second order statements if we reason in RCA^omega_0 in stead of RCA_0. However, the presence of finite types allows to treat various analytical notions in a rather direct way, compared to the encodings needed in RCA_0 which are not always provably faithful in RCA_0. Moreover, the language of finite types allows to treat many more principles and gives rise to interesting extensions of the existing scheme of reverse mathematics. We indicate this by showing that the class of principles equivalent (over RCA^omega_0 ) to Feferman's nonconstructive mu-operator forms a mathematically rich and very robust class. This is closely related to a phenomenon in higher type recursion theory known as Grilliot's trick.

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 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 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}.

scholarly journals Reverse Mathematics of  Divisibility in Integral  Domains

2021 ◽  
Author(s):  
◽  
Valentin B Bura
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
Ring Theory ◽  
The Other ◽  
Base System ◽  
Integral Domains ◽  
Second Order Arithmetic ◽  
Irreducible Factorization ◽  
Order Arithmetic ◽  
Well Foundedness

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

scholarly journals Countable sets versus sets that are countable in reverse mathematics

Computability ◽  
2021 ◽  
pp. 1-31
Author(s):  
Sam Sanders
Keyword(s):  
Reverse Mathematics ◽  
Higher Order ◽  
Second Order ◽  
Major Theme ◽  
Original Formulation ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Arithmetic ◽  
Definition Of

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic L 2 . A major theme in RM is therefore the study of structures that are countable or can be approximated by countable sets. Now, countable sets must be represented by sequences here, because the higher-order definition of ‘countable set’ involving injections/bijections to N cannot be directly expressed in L 2 . Working in Kohlenbach’s higher-order RM, we investigate various central theorems, e.g. those due to König, Ramsey, Bolzano, Weierstrass, and Borel, in their (often original) formulation involving the definition of ‘countable set’ based on injections/bijections to N. This study turns out to be closely related to the logical properties of the uncountably of R, recently developed by the author and Dag Normann. Now, ‘being countable’ can be expressed by the existence of an injection to N (Kunen) or the existence of a bijection to N (Hrbacek–Jech). The former (and not the latter) choice yields ‘explosive’ theorems, i.e. relatively weak statements that become much stronger when combined with discontinuous functionals, even up to Π 2 1 - CA 0 . Nonetheless, replacing ‘sequence’ by ‘countable set’ seriously reduces the first-order strength of these theorems, whatever the notion of ‘set’ used. Finally, we obtain ‘splittings’ involving e.g. lemmas by König and theorems from the RM zoo, showing that the latter are ‘a lot more tame’ when formulated with countable sets.

scholarly journals Reverse Mathematics of  Divisibility in Integral  Domains

2021 ◽  
Author(s):  
◽  
Valentin B Bura
Keyword(s):  
Reverse Mathematics ◽  
Second Order ◽  
Ring Theory ◽  
The Other ◽  
Base System ◽  
Integral Domains ◽  
Second Order Arithmetic ◽  
Irreducible Factorization ◽  
Order Arithmetic ◽  
Well Foundedness

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Hauptsatz for higher order logic

1968 ◽  
Vol 33 (3) ◽  
pp. 452-457 ◽  
Author(s):  
Dag Prawitz
Keyword(s):  
Finite Type ◽  
Type Theory ◽  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Simple Type ◽  
Higher Order Logic ◽  
Simple Type Theory ◽  
Second Order Logic

I shall prove in this paper that Gentzen's Hauptsatz is extendible to simple type theory, i.e., to the predicate logic obtained by admitting quantification over predicates of arbitrary finite type and generalizing the second order quantification rules to cover quantifiers of other types. (That Gentzen's Hauptsatz is extendible to this logic has been known as Takeuti's conjecture; see [4].) Gentzen's Hauptsatz has been extended to second order logic in a recent paper by Tait [3]. However, as remarked by Tait, his proof seems not to be extendible to higher orders. The present proof is rather an extension of a different proof of the Hauptsatz for second order logic that I have presented in [1].

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

scholarly journals The Brouwer invariance theorems in reverse mathematics

2020 ◽  
Vol 8 ◽  
Author(s):  
Takayuki Kihara
Keyword(s):  
Reverse Mathematics ◽  
Base System ◽  
Open Problems ◽  
Explicit Algorithm ◽  
Topological Embedding ◽  
Weak König’S Lemma ◽  
König’S Lemma

Abstract In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

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