An algorithm to determine, for any prime p, a polynomial-sized Horn sentence which expresses “The cardinality is not p”

Sauro Tulipani

doi:10.2307/2273991

An algorithm to determine, for any prime p, a polynomial-sized Horn sentence which expresses “The cardinality is not p”

Journal of Symbolic Logic ◽

10.2307/2273991 ◽

1985 ◽

Vol 50 (4) ◽

pp. 1062-1064 ◽

Cited By ~ 1

Author(s):

Sauro Tulipani

Keyword(s):

Horn Sentence

AbstractGiven a prime p, we exhibit a Horn sentence Hp which expresses “the cardinality is not p” and has size O(p5 log p).

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There are not exactly five objects

Journal of Symbolic Logic ◽

10.2307/2274177 ◽

1984 ◽

Vol 49 (2) ◽

pp. 467-469 ◽

Cited By ~ 2

Author(s):

Andreas Blass

Keyword(s):

Horn Sentence

AbstractWe exhibit a Horn sentence expressing the statement of the title; the construction generalizes to arbitrary primes in place of five.

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First Order Theory of Complete Stonean Algebras (Boolean-Valued Real and Complex Numbers)

Canadian Mathematical Bulletin ◽

10.4153/cmb-1987-057-7 ◽

1987 ◽

Vol 30 (4) ◽

pp. 385-392 ◽

Cited By ~ 3

Author(s):

Thomas Jech

Keyword(s):

Set Theory ◽

Order Theory ◽

Complex Numbers ◽

First Order ◽

First Order Theory ◽

Models Of Set Theory ◽

Horn Sentence

AbstractWe axiomatize the theory of real and complex numbers in Boolean-valued models of set theory, and prove that every Horn sentence true in the complex numbers is true in any complete Stonean algebra, and provable from its axioms.

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Logic of reduced power structures

Journal of Symbolic Logic ◽

10.2307/2273319 ◽

1983 ◽

Vol 48 (1) ◽

pp. 53-59

Author(s):

G.C. Nelson

Keyword(s):

Boolean Algebra ◽

Complete Theory ◽

Power Structures ◽

Index Set ◽

First Order ◽

Order Language ◽

Power Set ◽

Definition Of ◽

Proper Filter ◽

Horn Sentence

We start with the framework upon which this paper is based. The most useful reference for these notions is [2]. For any nonempty index set I and any proper filter D on S(I) (the power set of I), we denote by I/D the reduced power of modulo D as defined in [2, pp. 167–169]. The first-order language associated with I/D will always be the same language as associated with . We denote the 2-element Boolean algebra 〈{0, 1}, ⋂, ⋃, c, 0, 1〉 by 2 and 2I/D denotes the reduced power of it modulo D. We point out the intimate connection between the structures I/D and 2I/D given in [2, pp. 341–345]. Moreover, we assume as known the definition of Horn formula and Horn sentence as given in [2, p. 328] along with the fundamental theorem that φ is a reduced product sentence iff φ is provably equivalent to a Horn sentence [2, Theorem 6.2.5/ (iff φ is a 2-direct product sentence and a reduced power sentence [2, Proposition 6.2.6(ii)]). For a theory T(any set of sentences), ⊨ T denotes that is a model of T.In addition to the above we assume as known the elementary characteristics (due to Tarski) associated with a complete theory of a Boolean algebra, and we adopt the notation 〈n, p, q〉 of [3], [10], or [6] to denote such an elementary characteristic or the corresponding complete theory. We frequently will use Ershov's theorem which asserts that for each 〈n, p, q〉 there exist an index set I and filter D such that 2I/D ⊨ 〈n, p, q〉 [3] or [2, Lemma 6.3.21].

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