Omitting types for finite variable fragments and complete representations of algebras
Tarek Sayed Ahmed, Hajnal Andréka, and István Németi
Source: J. Symbolic Logic Volume 73, Issue 1
(2008), 65-89.
Abstract
We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n≥2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is ≥2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208358743
Digital Object Identifier: doi:10.2178/jsl/1208358743
Mathematical Reviews number (MathSciNet): MR2387933
Zentralblatt MATH identifier: 1143.03035
References
M. Amer, Cylindric algebras of sentences, Journal of Symbolic Logic, vol. 58 (1993), no. 2, p. 743.
M. Amer and T. Sayed Ahmed, Polyadic and cylindric algebras of sentences, Mathematical Logic Quarterly, vol. 52 (2006), no. 5, pp. 44--49.
Mathematical Reviews (MathSciNet): MR2268694
Digital Object Identifier: doi:10.1002/malq.200510039
Zentralblatt MATH: 1122.03062
H. Andréka, Atomic representable relation and cylindric algebras with non representable completions, Septembe r1997, manuscript, Rényi Institute of Mathematics, Budapest.
H. Andréka, S. Comer, J. X. Madarász, I. Németi, and T. Sayed Ahmed, Epimorphisms in cylindric algebras, Algebra Universalis, to appear, http://ftp.math-inst.hu/pub/algebraic-logic/ES -2007.pdf.
H. Andréka and I. Németi, Two-variable logic fails the omitting type property, 2007, manuscript, Rényi Mathematical Institute, Budapest.
H. Andréka, I. Németi, and I. Sain, Algebraic logic, Handbook of Philosophical Logic, vol. 2, Kluwer Academic Publishers, 2 ed., 2001, pp. 133--247.
Mathematical Reviews (MathSciNet): MR1884630
H. Andréka and T. Sayed Ahmed, Omitting types in logics with finitely many variables, Bulletin of Symbolic Logic, vol. 5 (1999), no. 1, p. 88.
H. Andréka, J. van Benthem, and I. Németi, Submodel preservation theorems in finite variable fragments, Modal Logic and Process Algebra. A Bisimulation Perspective (A. Ponse, M. de Rijke, and Y. Venema, editors), CSLI Lecture Notes, vol. 53, CSLI Publications, 1995, pp. 1--11.
Mathematical Reviews (MathSciNet): MR1375699
--------, Modal languages and bounded fragments of predicate logic, Journal of Philosophical Logic, vol. 27 (1998), pp. 217--274.
Mathematical Reviews (MathSciNet): MR1624137
Digital Object Identifier: doi:10.1023/A:1004275029985
Zentralblatt MATH: 0919.03013
B. Biró, Non-finite axiomatizability results in algebraic logic, Journal of Symbolic Logic, vol. 57 (1992), no. 3, pp. 832--843.
Mathematical Reviews (MathSciNet): MR1187451
Digital Object Identifier: doi:10.2307/2275434
Project Euclid: euclid.jsl/1183744043
Zentralblatt MATH: 0772.03032
C. Chang and J. Keisler, Model Theory, North Holland, 1994.
M. Ferenczi, On representability of neatly embeddable cylindric algebras, Journal of Applied Non-Classical Logics, vol. 3--4 (2000), pp. 300--315.
Mathematical Reviews (MathSciNet): MR1915688
--------, On representability of neatly embeddable cylindric algebras, Logic Journal of IGPL, in print.
E. Grädel, On the restraining power of guards, Journal of Symbolic Logic, vol. 64 (1999), no. 4, pp. 1719--1742.
Mathematical Reviews (MathSciNet): MR1780081
Digital Object Identifier: doi:10.2307/2586808
Project Euclid: euclid.jsl/1183745949
Zentralblatt MATH: 0958.03027
E. Grädel and R. Rosen, On preservation theorems for two-variable logic, Mathematical Logic Quarterly, vol. 45 (1999), pp. 315--325.
Mathematical Reviews (MathSciNet): MR1703717
L. Henkin, A generalization of the concept of $\omega$-consistency, Journal of Symbolic Logic, vol. 19 (1954), pp. 183--196.
Mathematical Reviews (MathSciNet): MR63324
Digital Object Identifier: doi:10.2307/2268617
Project Euclid: euclid.jsl/1183731924
Zentralblatt MATH: 0056.01104
--------, The representation theorem for cylindric algebras, Mathematical Interpretation of Formal Systems, North-Holland, Amsterdam, 1955, pp. 85--97.
Mathematical Reviews (MathSciNet): MR75155
Zentralblatt MATH: 0087.25001
--------, Logical systems containing only a finite number of symbols, Séminaire de Mathématiques Supérieures, vol. 21, Les Presses de l'Université de Montréal, 1967.
Mathematical Reviews (MathSciNet): MR239959
Zentralblatt MATH: 0164.30702
L. Henkin, J. D. Monk, and A. Tarski, Cylindric Algebras Parts I, II, North-Holland, 1971, 1985.
Mathematical Reviews (MathSciNet): MR781929
Zentralblatt MATH: 0576.03042
R. Hirsch and I. Hodkinson, Complete representations in algebraic logic, Journal of Symbolic Logic, vol. 62 (1997), no. 3, pp. 816--847.
Mathematical Reviews (MathSciNet): MR1472125
Digital Object Identifier: doi:10.2307/2275574
Project Euclid: euclid.jsl/1183745299
Zentralblatt MATH: 0893.03025
--------, Relation algebras from cylindric algebras I, Journal of Pure and Applied Logic, vol. 112 (2001), pp. 225--266.
Mathematical Reviews (MathSciNet): MR1870968
Digital Object Identifier: doi:10.1016/S0168-0072(01)00084-7
Zentralblatt MATH: 1001.03057
--------, Relation algebras from cylindric algebras II, Journal of Pure and Applied Logic, vol. 112 (2001), pp. 267--297.
Mathematical Reviews (MathSciNet): MR1870969
Digital Object Identifier: doi:10.1016/S0168-0072(01)00085-9
Zentralblatt MATH: 1001.03058
--------, Relation Algebras by Games, Studies in Logic and Foundations of Mathematics, vol. 147, North-Holland, Amsterdam, 2002.
Mathematical Reviews (MathSciNet): MR1935083
Zentralblatt MATH: 1018.03002
--------, Strongly representable atom structures of relation algebras, Proceedings of the American Mathematican Society, vol. 130 (2002), pp. 1819--1831.
Mathematical Reviews (MathSciNet): MR1887031
Digital Object Identifier: doi:10.1090/S0002-9939-01-06232-3
JSTOR: links.jstor.org
Zentralblatt MATH: 1002.03054
R. Hirsch, I. Hodkinson, and R. Maddux, On provability with finitely many variables, Bulletin of Symbolic Logic, vol. 8 (2002), pp. 348--379.
Mathematical Reviews (MathSciNet): MR1931348
Project Euclid: euclid.bsl/1182353893
--------, Relation algebra reducts of cylindric algebras and an application to proof theory, Journal of Symbolic Logic, vol. 67 (2002), no. 1, pp. 197--213.
Mathematical Reviews (MathSciNet): MR1889544
Digital Object Identifier: doi:10.2178/jsl/1190150037
Project Euclid: euclid.jsl/1190150037
Zentralblatt MATH: 1005.03052
I. Hodkinson, Atom structures of cylindric algebras and relation algebras, Annals of Pure and Applied Logic, vol. 89 (1997), pp. 117--148.
Mathematical Reviews (MathSciNet): MR1490103
Digital Object Identifier: doi:10.1016/S0168-0072(97)00015-8
Zentralblatt MATH: 0898.03025
E. Hoogland and M. Marx, Interpolation in the guarded fragment, Studia Logica, vol. 70 (2002), no. 3, pp. 273--409.
Mathematical Reviews (MathSciNet): MR1897997
Digital Object Identifier: doi:10.1023/A:1015154431342
Zentralblatt MATH: 0998.03004
E. Hoogland, M. Marx, and M. Otto, Beth definability for the guarded fragment, Proceedings of 6th International Conference on Logic Programming and Automated Reasoning (LPAR), Tbilisi, Georgia, Lecture Notes in Computer Science, vol. 1705, Springer Verlag, 1999, pp. 273--285.
Mathematical Reviews (MathSciNet): MR1853176
Zentralblatt MATH: 0939.03038
Digital Object Identifier: doi:10.1007/3-540-48242-3_17
B. Jónsson and A. Tarski, Boolean algebras with operators, Part I, American Journal of Mathematics, vol. 73 (1951), pp. 891--939.
Mathematical Reviews (MathSciNet): MR44502
Digital Object Identifier: doi:10.2307/2372123
JSTOR: links.jstor.org
Zentralblatt MATH: 0045.31505
J. X. Madarász, Logic and relativity (in the light of definability theory), Ph.D. thesis, Budapest, 2002, available at http://www.math-inst.hu/pub/algebraic-logic/diszi0226.pdf.gz.
J. X. Madarász and T. Sayed Ahmed, Amalgamation, interpolation and epimorphisms in algebraic logic, Algebra Universalis, vol. 56 (2007), no. 2, pp. 179--210.
Mathematical Reviews (MathSciNet): MR2287826
Digital Object Identifier: doi:10.1007/s00012-007-1987-3
Zentralblatt MATH: 1122.03063
R. Maddux, Non-finite axiomatizability results for cylindric and relational algebras, Journal of Symbolic Logic, vol. 54 (1989), no. 3, pp. 951--974.
Mathematical Reviews (MathSciNet): MR1011183
Digital Object Identifier: doi:10.2307/2274756
Project Euclid: euclid.jsl/1183743031
Zentralblatt MATH: 0686.03035
--------, The neat embedding problem and the number of variables in proofs, Proceedings of the American Mathematical Society, vol. 112 (1991), pp. 195--202.
Mathematical Reviews (MathSciNet): MR1033959
Digital Object Identifier: doi:10.2307/2048497
JSTOR: links.jstor.org
Zentralblatt MATH: 0717.03025
J. D. Monk, Non-finitizability of classes of representable cylindric algebras, Journal of Symbolic Logic, vol. 34 (1969), no. 3, pp. 331--343.
Mathematical Reviews (MathSciNet): MR256861
Digital Object Identifier: doi:10.2307/2270900
Project Euclid: euclid.jsl/1183736847
Zentralblatt MATH: 0181.30002
--------, Completions of boolean algebras with operators, Mathematische Nachrichten, vol. 46 (1970), pp. 47--55.
Mathematical Reviews (MathSciNet): MR277369
Digital Object Identifier: doi:10.1002/mana.19700460105
Zentralblatt MATH: 0182.32301
D. Mundici, Complexity of Craig's interpolation, Fundamenta Informaticae, vol. 5 (1982), pp. 261--278.
Mathematical Reviews (MathSciNet): MR697483
I. Németi, Algebraisation of quantifier logics, an introductory overview, Studia Logica, vol. 50 (1991), no. 4, pp. 465--569, Updated version with proofs is Mathematical Institute Budapest Preprint No 13-1996. Electronically available at http://www.math-inst.hu/pub/algebraic-logic/survey.ps.
Mathematical Reviews (MathSciNet): MR1170186
Digital Object Identifier: doi:10.1007/BF00370684
Zentralblatt MATH: 0772.03033
S. Orey, On $omega$-consistency and related properties, Journal of Symbolic Logic, vol. 21 (1956), pp. 246--252.
Mathematical Reviews (MathSciNet): MR82936
Digital Object Identifier: doi:10.2307/2269096
Project Euclid: euclid.jsl/1183732456
Zentralblatt MATH: 0071.00801
G. Sági, A note on algebras of substitution, Studia Logica, vol. 72 (2002), no. 2, pp. 265--284.
Mathematical Reviews (MathSciNet): MR1949916
Digital Object Identifier: doi:10.1023/A:1021364629235
Zentralblatt MATH: 1010.03053
I. Sain, On the search for a finitizable algebraization of first-order logic, Logic Journal of IGPL, vol. 8 (2000), no. 4, pp. 497--591.
Mathematical Reviews (MathSciNet): MR1776151
Digital Object Identifier: doi:10.1093/jigpal/8.4.497
Zentralblatt MATH: 0973.03008
I. Sain and V. Gyuris, Finite schematizable algebraic logic, Logic Journal of the IGPL, vol. 5 (1997), no. 5, pp. 699--751.
Mathematical Reviews (MathSciNet): MR1465620
Digital Object Identifier: doi:10.1093/jigpal/5.5.699
Zentralblatt MATH: 0886.03044
T. Sayed Ahmed, Martin's axiom, omitting types and complete representations in algebraic logic, Studia Logica, vol. 72 (2002), pp. 1--25.
Mathematical Reviews (MathSciNet): MR1949917
Digital Object Identifier: doi:10.1023/A:1021368713305
Zentralblatt MATH: 1019.03047
--------, Topics in algebraic logic (on neat reducts of algebras of logic), Ph.D. thesis, Cairo University, Cairo Egypt, 2002.
--------, Neat embeddings, omitting types and interpolation, Notre Dame Journal of Formal Logic, vol. 44 (2003), no. 3, pp. 157--173.
Mathematical Reviews (MathSciNet): MR2130788
Digital Object Identifier: doi:10.1305/ndjfl/1091030854
Project Euclid: euclid.ndjfl/1091030854
Zentralblatt MATH: 1071.03041
--------, On amalgamation of reducts of polyadic algebras, Algebra Universalis, vol. 51 (2004), pp. 301--359, preprint available at http://www.math-inst.hu/pub/algebraic-logic.
Mathematical Reviews (MathSciNet): MR2082131
Digital Object Identifier: doi:10.1007/s00012-004-1807-y
Zentralblatt MATH: 1126.03053
--------, Algebraic logic, where does it stand today?, Bulletin of Symbolic Logic, vol. 11 (2005), no. 4, pp. 465--516.
Mathematical Reviews (MathSciNet): MR2198710
Digital Object Identifier: doi:10.2178/bsl/1130335206
Project Euclid: euclid.bsl/1130335206
Zentralblatt MATH: 1111.03053
--------, Omitting types for algebraizable extensions of first order logic, Journal of Applied Non-classical Logics, vol. 15 (2006), no. 4, pp. 465--490.
Mathematical Reviews (MathSciNet): MR2301390
Digital Object Identifier: doi:10.3166/jancl.15.465-489
Zentralblatt MATH: 05640519
T. Sayed Ahmed and I. Németi, On neat reducts of algebras of logic, Studia Logica, vol. 68 (2001), pp. 229--262.
Mathematical Reviews (MathSciNet): MR1860732
Digital Object Identifier: doi:10.1023/A:1012447223176
Zentralblatt MATH: 0993.03085
A. Simon, Connections between quasi-projective relation algebras and cylindric algebras, Algebra Universalis, vol. 56 (2007), no. 3--4, pp. 263--301.
Mathematical Reviews (MathSciNet): MR2318212
Digital Object Identifier: doi:10.1007/s00012-007-1999-z
Zentralblatt MATH: 1135.03029
A. Tarski, Logic, Semantics, Metamathematics: Papers from 1923 to 1938, Clarendon Press, Oxford, 1956, Translated by J. H. Woodger.
Mathematical Reviews (MathSciNet): MR78296
A. Tarski and S. R. Givant, A formalization of set theory without variables, Colloquium of Publications of the American Mathematical Society, vol. 41, Rhode Island, 1987.
