The Baire category theorem in weak subsystems of second-order arithmetic

1993 ◽  
Vol 58 (2) ◽  
pp. 557-578 ◽  
Author(s):  
Douglas K. Brown ◽  
Stephen G. Simpson
Keyword(s):  
Functional Analysis ◽  
Model Theory ◽  
Baire Category ◽  
Second Order ◽  
Base System ◽  
Baire Category Theorem ◽  
Second Order Arithmetic ◽  
Category Theorem ◽  
Order Arithmetic

AbstractWorking within weak subsystems of second-order arithmetic Z2 we consider two versions of the Baire Category theorem which are not equivalent over the base system RCA0. We show that one version (B.C.T.I) is provable in RCA0 while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of Z2, which we call and , and , show that suffices to prove B.C.T.II. Some model theory of and its importance in view of Hilbert's program is discussed, as well as applications of our results to functional analysis.

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.

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 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 Short note: Least fixed points versus least closed points

2021 ◽  
Author(s):  
Gerhard Jäger
Keyword(s):  
Fixed Points ◽  
Short Note ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Arithmetic Operator

AbstractThis short note is on the question whether the intersection of all fixed points of a positive arithmetic operator and the intersection of all its closed points can proved to be equivalent in a weak fragment of second order arithmetic.

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

2014 ◽  
Vol 79 (4) ◽  
pp. 1001-1019 ◽  
Author(s):  
ASHER M. KACH ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Direct Product ◽  
Disjoint Union ◽  
Second Order ◽  
Linear Orders ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Natural Classes ◽  
Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

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