Effectively inseparable Boolean algebras in lattices of sentences

Shavrukov, V. Yu.

doi:10.1007/s00153-009-0161-3

Cited by 4 publications

(8 citation statements)

References 12 publications

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“…For introductions to e.i. Boolean pre-algebras and related topics, see (listed in chronological order) [18,17,19,26].…”

Section: Examples Of Ei Pre-latticesmentioning

confidence: 99%

“…Shavrukov [26] proved that for n ě 1 the c.e. Boolean pre-algebra B ∆n{T (having as domain the ∆ n -sentences, and regarded as a c.e.…”

Section: 2mentioning

confidence: 99%

“…Boolean pre-algebras" are e.i. Boolean algebras in the sense of [18,26]. See also [23,11,12,9] for an approach to c.e structures similar to the one taken in this paper.…”

Section: Introductionmentioning

confidence: 95%

“…Boolean prealgebras" are e.i. Boolean algebras in the sense of (Nerode & Remmel, 1985;Shavrukov, 2010). See also (Fokina, Khoussainov, Semukhin, & Turetsky, 2016;Gavryushkin, Jain, Khoussainov, & Stephan, 2014;Gavryushkin, Khoussainov, & Stephan, 2016;Selivanov, 2003) for an approach to c.e structures similar to the one taken in this article.…”

mentioning

confidence: 99%

“…preordering relations is based on the notion of effective inseparability for lattices (or, better, prelattices, as we will call them in this article). In this regard, this article can be viewed as a generalization of some aspects of the work in (Montagna & Sorbi, 1985;Nies, 2000;Shavrukov, 2010), concerning effectively inseparable Boolean algebras.…”

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confidence: 99%

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Effective Inseparability, Lattices, and Preordering Relations

Andrews¹,

Sorbi²

2019

The Review of Symbolic Logic

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We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form L " xω,^, _, 0, 1, ďLy where ω denotes the set of natural numbers and the following hold:^, _ are binary computable operations; ďL is a c.e. pre-ordering relation, with 0 ďL x ďL 1 for every x; the equivalence relation "L originated by ďL is a congruence on L such that the corresponding quotient structure is a non-trivial bounded lattice; the "L-equivalence classes of 0 and 1 form an effectively inseparable pair), and show (Theorem 30, solving a problem in [17]), that if L is an e.i. pre-lattice then ďL is universal with respect to all c.e. pre-ordering relations, i.e. for every c.e. pre-ordering relation R there exists a computable function f such that, for all x, y, x R y if and only if f pxq ďL f pyq; in fact (Corollary 43) ďL is locally universal, i.e. for every pair a ăL b and every c.e. pre-ordering relation R one can find a reducing function f from R to ďL such that the range of f is contained in the interval tx : a ďL x ďL bu. Also (Theorem 47) ďL is uniformly dense, i.e. there exists a computable function f such that for every a, b if a ăL b then a ăL f pa, bq ăL b, and if a "L a 1 and b "L b 1 then f pa, bq "L f pa 1 , b 1 q. Some consequences and applications of these results are discussed: in particular (Corollary 55) for n ě 1 the c.e. pre-ordering relation on Σn sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson's Q or R is locally universal and uniformly dense; and (Corollary 56) the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.2010 Mathematics Subject Classification. 03D25.

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“…For introductions to e.i. Boolean pre-algebras and related topics, see (listed in chronological order) [18,17,19,26].…”

Section: Examples Of Ei Pre-latticesmentioning

confidence: 99%

“…Shavrukov [26] proved that for n ě 1 the c.e. Boolean pre-algebra B ∆n{T (having as domain the ∆ n -sentences, and regarded as a c.e.…”

Section: 2mentioning

confidence: 99%

“…Boolean pre-algebras" are e.i. Boolean algebras in the sense of [18,26]. See also [23,11,12,9] for an approach to c.e structures similar to the one taken in this paper.…”

Section: Introductionmentioning

confidence: 95%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Effective Inseparability, Lattices, and Preordering Relations

Andrews¹,

Sorbi²

2019

The Review of Symbolic Logic

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Interpretability degrees of finitely axiomatized sequential theories

Visser

2013

Arch. Math. Logic

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Abstract. In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory -like Elementary Arithmetic EA, IΣ 1 , or the Gödel-Bernays theory of sets and classes GB-have suprema. This partially answers a question posed by VítěslavŠvejdar in his paper [Šve78]. The partial solution ofŠvejdar's problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree structure of the degrees of all finitely axiomatized sequential theories.In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result.Dedicated to Dirk van Dalen on the occasion of his 80th birthday.

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Franco Montagna’s Work on Provability Logic and Many-valued Logic

Beklemishev

Flaminio

2016

Stud Logica

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Effectively inseparable Boolean algebras in lattices of sentences

Cited by 4 publications

References 12 publications

Effective Inseparability, Lattices, and Preordering Relations

Effective Inseparability, Lattices, and Preordering Relations

Interpretability degrees of finitely axiomatized sequential theories

Franco Montagna’s Work on Provability Logic and Many-valued Logic

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