Omitting types for algebraizable extensions of first order logic

We prove an Omitting Types Theorem for certain algebraizable extensions of first order logic without equality studied in [SAI 00] and [SAY 04]. This is done by proving a representation theorem preserving given countable sets of infinite meets for certain reducts of ω- dimensional polyadic algebras, the so-called G polyadic algebras (Theorem 5). Here G is a special subsemigroup of (ω, ω o) that specifies the signature of the algebras in question. We state and prove an independence result connecting our representation theorem to Martin's axiom (Theorem 6). Also we show that the countable atomic G polyadic algebras are completely representable (Corollary 16) contrasting results on cylindric algebras. Several related results are surveyed.

[1]  András Simon,et al.  What the finitization problem is not , 1993 .

[2]  Tarek Sayed Ahmed,et al.  Algebraic Logic, Where Does it Stand Today? , 2005, Bulletin of Symbolic Logic.

[3]  T. Gergely,et al.  On universal algebraic constructions of logics , 1977 .

[4]  Robert Goldblatt,et al.  Persistence and Atomic Generation for Varieties of Boolean Algebras with Operators , 2001, Stud Logica.

[5]  István Németi.  Strong Representability of Fork Algebras, a Set Theoretic Foundation , 1997, Log. J. IGPL.

[6]  A. Tarski,et al.  A Formalization Of Set Theory Without Variables , 1987 .

[7]  Arnold W. Miller,et al.  Some properties of measure and category , 1981 .

[8]  Balázs Biró.  Isomorphic but Not Lower Base-Isomorphic Cylindric Algebras of Finite Dimension , 1989, Notre Dame J. Formal Log..

[9]  Robert Goldblatt,et al.  On the role of the Baire Category Theorem and Dependent Choice in the foundations of logic , 1985, Journal of Symbolic Logic.

[10]  Roger D. Maddux,et al.  Nonfinite axiomatizability results for cylindric and relation algebras , 1989, Journal of Symbolic Logic.

[11]  Hajnal Andréka,et al.  Complexity of Equations Valid in Algebras of Relations: Part I: Strong Non-Finitizability , 1997, Ann. Pure Appl. Log..

[12]  A. Tarski,et al.  Cylindric Algebras. Part II , 1988 .

[13]  James S. Johnson,et al.  Nonfinitizability of classes of representable polyadic algebras , 1969, Journal of Symbolic Logic.

[14]  Balázs Biró.  Non-Finite-Axiomatizability Results in Algebraic Logic , 1992, J. Symb. Log..

[15]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[16]  G. Sági.  NON-COMPUTABILITY OF THE EQUATIONAL THEORY OF POLYADIC ALGEBRAS , 2001 .

[17]  Tarek Sayed Ahmed,et al.  On amalgamation of reducts of polyadic algebras , 2004 .

[18]  Ludomir Newelski.  Omitting Types and the Real Line , 1987, J. Symb. Log..

[19]  Y. Venema.  Atom structures and Sahlqvist equations , 1997 .

[20]  Larisa Maksimova,et al.  Amalgamation and interpolation in normal modal logics , 1991, Stud Logica.

[21]  Judit X. Madarász,et al.  Interpolation in Algebraizable Logics; Semantics for Non-normal Multi-modal Logic , 1998, J. Appl. Non Class. Logics.

[22]  Ildikó Sain,et al.  Finite Schematizable Algebraic Logic , 1997, Log. J. IGPL.

[23]  Saharon Shelah,et al.  Isomorphic but not Lower Base-Isomorphic Cylindric Set Algebras , 1988, J. Symb. Log..

[24]  István Németi,et al.  On the equational theory of representable polyadic equality algebras , 2000, Journal of Symbolic Logic.

[25]  Enrique Casanovas,et al.  Omitting Types in Incomplete Theories , 1996, J. Symb. Log..

[26]  Ian M. Hodkinson,et al.  Atom Structures of Cylindric Algebras and Relation Algebras , 1997, Ann. Pure Appl. Log..

[27]  Ian M. Hodkinson,et al.  Complete representations in algebraic logic , 1997, Journal of Symbolic Logic.

[28]  Tarek Sayed Ahmed,et al.  Martin's Axiom, Omitting Types, and Complete Representations in Algebraic Logic , 2002, Stud Logica.

[29]  Ian M. Hodkinson,et al.  Step by step – Building representations in algebraic logic , 1997, Journal of Symbolic Logic.

[30]  Tarek Sayed Ahmed,et al.  On Neat Reducts of Algebras of Logic , 2001, Stud Logica.

[31]  Azriel Levy.  Basic set theory , 1979 .

[32]  Ildikó Sain,et al.  On the Search for a Finitizable Algebraization of First Order Logic , 2000, Log. J. IGPL.

[33]  Gábor Sági.  A Completeness Theorem for Higher Order Logics , 2000, J. Symb. Log..

[34]  J. Donald Monk,et al.  Nonfinitizability of Classes of Representable Cylindric Algebras , 1969, J. Symb. Log..