Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?

1984 ◽  
Vol 49 (3) ◽  
pp. 783-802 ◽  
Author(s):  
Stephen G. Simpson
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Formal System ◽  
Computable Analysis ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Weak König’S Lemma ◽  
König’S Lemma ◽  
Degrees Of Unsolvability

AbstractWe investigate the provability or nonprovability of certain ordinary mathematical theorems within certain weak subsystems of second order arithmetic. Specifically, we consider the Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the context of the formal system RCA0 whose principal axioms are comprehension and induction. Our main result is that, over RCA0, the Cauchy/Peano Theorem is provably equivalent to weak König's lemma, i.e. the statement that every infinite {0, 1}-tree has a path. We also show that, over RCA0, the Ascoli lemma is provably equivalent to arithmetical comprehension, as is Osgood's theorem on the existence of maximum solutions. At the end of the paper we digress to relate our results to degrees of unsolvability and to computable analysis.

scholarly journals THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

2017 ◽  
Vol 10 (2) ◽  
pp. 357-396 ◽  
Author(s):  
WALTER DEAN ◽  
SEAN WALSH
Keyword(s):  
Large Scale ◽  
Second Order ◽  
Descriptive Set Theory ◽  
Second Order Arithmetic ◽  
Mathematics Program ◽  
Order Arithmetic ◽  
Transfinite Recursion ◽  
Descriptive Set ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractThis paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König’s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

scholarly journals Finding paths through narrow and wide trees

2009 ◽  
Vol 74 (1) ◽  
pp. 349-360 ◽  
Author(s):  
Stephen Binns ◽  
Bjørn Kjos-Hanssen
Keyword(s):  
Second Order ◽  
Binary Trees ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Weak König’S Lemma ◽  
König’S Lemma ◽  
Dual Statement

AbstractWe consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Aldona Dutkiewicz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Kuratowski Measure Of Noncompactness ◽  
Japan Acad

AbstractIn this paper we prove an existence theorem for ordinary differential equations in Banach spaces. The main assumptions in our results, formulated in terms of the Kuratowski measure of noncompactness, are motivated by the paper [CONSTANTIN, A.: On Nagumo’s theorem, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 41-44].

scholarly journals An existence theorem for ordinary differential equations in Banach spaces

1984 ◽  
Vol 30 (3) ◽  
pp. 449-456 ◽  
Author(s):  
Bogdan Rzepecki
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Regularity Condition ◽  
Measure Of Noncompactness ◽  
Bounded Solution ◽  
Carathéodory Conditions

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

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