On Compactness of Logics That Can Express Properties of Symmetry or Connectivity

A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic idea underlying the results and examples presented here is that it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset T′ of T has a model which is rigid.

[1]  Jon Barwise,et al.  A correction to “stationary logic” , 1981 .

[2]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[3]  Richard Kaye,et al.  Automorphisms of first-order structures , 1994 .

[4]  Nicholas C. Wormald,et al.  The asymptotic connectivity of labelled regular graphs , 1981, J. Comb. Theory B.

[5]  Vera Koponen,et al.  Random graphs with bounded maximum degree: asymptotic structure and a logical limit law , 2012, Discret. Math. Theor. Comput. Sci..

[6]  S. Shelah.  Generalized quantifiers and compact logic , 1975 .

[7]  H. Keisler.  Logic with the quantifier “there exist uncountably many” , 1970 .

[8]  SAMEER KAILASA,et al.  TOPICS IN GEOMETRIC GROUP THEORY , 2015 .

[9]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[10]  D. Marker.  Model theory : an introduction , 2002 .

[11]  R. L. Vaught,et al.  The completeness of logic with the added quantifier "there are uncountable many" , 1964 .

[12]  Nicholas C. Wormald,et al.  The asymptotic distribution of short cycles in random regular graphs , 1981, J. Comb. Theory, Ser. B.

[13]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[14]  Gebhard Fuhrken.  LANGUAGES WITH ADDED QUANTIFIER “THERE EXIST AT LEAST χα” , 2014 .

[15]  Jörg Flum,et al.  Finite model theory , 1995, Perspectives in Mathematical Logic.

[16]  Jon Barwise,et al.  Model-Theoretic Logics , 2016 .

[18]  David Marker,et al.  Introduction to Model Theory , 2000 .

[19]  Per Lindström,et al.  On Extensions of Elementary Logic , 2008 .

[20]  Brendan D. McKay,et al.  Automorphisms of random graphs with specified vertices , 1984, Comb..

[21]  Jouko A. Väänänen,et al.  Barwise: Abstract Model Theory and Generalized Quantifiers , 2004, Bulletin of Symbolic Logic.

[22]  D. Evans.  Model Theory of Groups and Automorphism Groups , 1997 .

[23]  P. Rothmaler.  Introduction to Model Theory , 2000 .

[24]  La Harpe,et al.  Topics in Geometric Group Theory , 2000 .