Addendum to “countable algebra and set existence axioms”

Friedman, Harvey M.; Simpson, Stephen G.; Smith, Rick L.

doi:10.1016/0168-0072(85)90020-x

Cited by 53 publications

(94 citation statements)

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“…the reverse mathematician is constrained to use full induction as sparingly as possible" [Simpson, 1985, p. 150]. It is additionally observed in both [Friedman et al, 1983] and [Simpson, 1985] that the systems with restricted induction are more amenable to ordinal analysis, that their firstorder parts of systems with restricted induction typically admit neater characterizations, and that while the use of restricted induction sometimes results in more complicated proofs, not only are few reversals to classical theorems lost in this manner, but their proofs sometimes yields improved quantifiers bounds.…”

Section: Rca 0 As a Formal Systemmentioning

confidence: 99%

“…But although his original formulation of the basic subsystems in [Friedman, 1975] was based on full first-order induction, no explanation for restricting induction is given in [Friedman, 1976]. This topic is, however, discussed at length in [Friedman et al, 1983] wherein RCA 0 appears to have first been formulated in its contemporary form. One programmatic observation made there is that full induction is provably equivalent to the bounded comprehension scheme ∀x∃X(y ∈ X ↔ (y < x ∧ φ(y))) over RCA 0 .…”

Section: Rca 0 As a Formal Systemmentioning

confidence: 99%

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The Prehistory of the Subsystems of Second-Order Arithmetic

Dean

Walsh

2017

The Review of Symbolic Logic

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This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of 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.

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Section: Rca 0 As a Formal Systemmentioning

confidence: 99%

Section: Rca 0 As a Formal Systemmentioning

confidence: 99%

The Prehistory of the Subsystems of Second-Order Arithmetic

Dean

Walsh

2017

The Review of Symbolic Logic

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“…The ongoing study has been carried out in the context of subsystems of second order arithmetic, under the slogan Reverse Mathematics [12,3,23]. We continue this program here by examining the role of strong set existence axioms in separable Banach space theory.…”

Section: Introductionmentioning

confidence: 99%

“…By Bounded König's Lemma, T f has a path. Here we are using the fact that Bounded König's Lemma is provable in WKL 0 (see [12] and Lemma IV.1.4 of [23]). Since T f ⊆ T , it follows that T has a path.…”

Section: Be the Upward Closure Of T Under Majorization Then T Is Welmentioning

confidence: 99%

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Separable Banach space theory needs strong set existence axioms

Humphreys

Simpson

1996

Trans. Amer. Math. Soc.

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Abstract. We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π 1 1 comprehension, is needed to prove such basic facts as the existence of the weak- * closure of any norm-closed subspace of 1 = c * 0 . This is in contrast to earlier work [6,4, 7,23,22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π 0 2 sentences. En route to our main results, we prove the Krein-Šmulian theorem in ACA 0 , and we give a new, elementary proof of a result of McGehee on weak- * sequential closure ordinals.

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Orders on computable rings

2020

Mathematical Logic Qtrly

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The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to WKL0 over RCA0. We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to WKL0 over RCA0. In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre‐orders to orders, where Zorn's lemma is used. We then prove that “pre‐orders on rings not necessarily commutative extend to orders” is equivalent to WKL0.

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Addendum to “countable algebra and set existence axioms”

Abstract: 2016-12-26T14:59:35

Cited by 53 publications

References 1 publication

The Prehistory of the Subsystems of Second-Order Arithmetic

The Prehistory of the Subsystems of Second-Order Arithmetic

Separable Banach space theory needs strong set existence axioms

Orders on computable rings

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