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Algebra and Logic

, Volume 57, Issue 5, pp 368–380 | Cite as

Forcing Formulas in Fraïssé Structures and Classes

  • A. T. Nurtazin
Article
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We come up with a semantic method of forcing formulas by finite structures in an arbitrary fixed Fraïssé class Open image in new window . Both known and some new necessary and sufficient conditions are derived under which a given structure Open image in new window will be a forcing structure. A formula φ is forced on \( \overline{a} \) in an infinite structure Open image in new window ╟φ\( \left(\overline{a}\right) \) if it is forced in Open image in new window by some finite substructure of Open image in new window . It is proved that every ∃∀∃-sentence true in a forcing structure is also true in any existentially closed companion of the structure. The new concept of a forcing type plays an important role in studying forcing models. It is proved that an arbitrary structure will be a forcing structure iff all existential types realized in the structure are forcing types. It turns out that an existentially closed structure which is simple over a tuple realizing a forcing type will itself be a forcing structure. Moreover, every forcing type is realized in an existentially closed structure that is a model of a complete theory of its forcing companion.

Keywords

forcing method Fraïssé class forcing structure forcing type existentially closed structure existentially closed companion 

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References

  1. 1.
    A. T. Nurtazin, “Countable infinite existentially closed models of universally axiomatizable theories,” Sib. Adv. Math., 26, No. 2, 99-125 (2016).MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. J. Cohen, Set Theory and the Continuum Hypothesis, W. A. Benjamin, New York (1966).Google Scholar
  3. 3.
    J. Barwise and A. Robinson, “Completing theories by forcing,” Ann. Math. Log., 2, 119-142 (1970).MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Fraïssé, “Sur quelques classifications des systèmes de rélations,” Publ. Sci. Univ. Alger, Sér. A, 1, 35-182 (1955).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Handbook of Mathematical Logic, Vol. 1, Model Theory, J. Barwise (ed.), North-Holland, Amsterdam (1977).Google Scholar
  6. 6.
    A. T. Nurtazin “Properties of existentially closed companions,” Algebra and Logic, 57, No. 3, 211-221 (2018).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Information and Computational Technologies, Ministry of Education and Science RKAlma-AtaKazakhstan

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