\(\Pi_1^1\)-CA\(_0\) and order types of countable ordered groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 0981.03060
DOI: 10.2307/2694917
publication date: 2001-08-01T00:00:00Z
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 1623517
zbMATH Keywords: reverse mathematics
zbMATH Keywords: order types
zbMATH Keywords: ordered groups
author: Reed Solomon
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2694917
OpenAlex ID: W4231591585
cites work: Countable algebra and set existence axioms
cites work: Weak comparability of well orderings and reverse mathematics
cites work: Ordered Groups: A Case Study in Reverse Mathematics
cites work: Q4220572
cites work: Ordinal inequalities, transfinite induction, and reverse mathematics
cites work: Set theory. An introduction to independence proofs
cites work: Degrees of orderings not isomorphic to recursive linear orderings
Recommended article: Q3464625
Recommended article: Q5711887
Recommended article: Ordered Groups: A Case Study in Reverse Mathematics
Recommended article: Reverse Mathematics and Ordinal Multiplication
Recommended article: Q3699664
Recommended article: Invariants, Boolean algebras and ACA₀⁺
Recommended article: Q5711881
Recommended article: Q5711888
Recommended article: Ranked structures and arithmetic transfinite recursion
Recommended article: Reverse mathematics and fully ordered groups
title: \(\Pi_1^1\)-CA\(_0\) and order types of countable ordered groups
published in: The Journal of Symbolic Logic
review text: The author deals with the classification problem of all admissible order types for orderable groups from the point of view of the Friedman-Simpson program of reverse mathematics [cf. \textit{S. G. Simpson}, ``Reverse mathematics'', Proc. Symp. Pure Math. 42, 461-471 (1985; Zbl 0578.03005), Subsystems of second order arithmetic, Springer, Berlin-Heidelberg (1999; Zbl 0909.03048)]. It is well known that in the abelian case a basic tool is the classical theorem of Hölder (stating that every archimedean ordered group is isomorphic to a subgroup of the additive ordered group of the reals), which can be proven in the weakest system of reverse mathematics, namely recursive analysis \(\text{RCA}_0\) [cf. \textit{R. Solomon}, ``Ordered groups: a case study in reverse mathematics'', Bull. Symb. Log. 5, No.~1, 45-58 (1999; Zbl 0922.03078)]. In the case of countable ordered groups, the fundamental classification theorem is due to Mal'tsev: if \(G\) is a countable ordered group, then the order type of \(G\) has the form \(Z^\alpha Q^\varepsilon\), for some wellorder \(\alpha\) and \(\varepsilon \in \{0,1\}\) (where \(Z\) is the additive group of the integers and \(Q\) the field of the rationals). The main content of this paper is the proof that Mal'tsev 's theorem is indeed much stronger, and in fact equivalent with the strongest subsystem investigated so far in reverse mathematics, namely \(\Pi ^1_1\text{-CA}_0\). Therefore, Mal'tsev's theorem is not predicatively reducible (e.g. provable in the system based on arithmetical transfinite recursion \(\text{ATR}_0\)). We recall that \(\Pi ^1_1\text{-CA}_0\) is based on the schema postulating the existence of a set of natural numbers in correspondence to arbitrary formulas of the form \(\forall X A(X,u)\), \(A\) being first order (no set quantifier). The result should be especially appreciated, since, as the reader can check from Simpson's book, the number of known equivalences of mathematical results with the subsystem \(\Pi ^1_1\t…
reviewed by: Andrea Cantini
description: scientific article; zbMATH DE number 1623517