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On the equational class of diagonalizable algebras

The algebraization of the theories which express Theor; VI

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Summary

It is well-known that, in Peano arithmetic, there exists a formulaTheor(x) which numerates the set of theorems and that this formula satisfies Hilbert-Bernays derivability conditions. RecentlyR. Magari has suggested an algebraization of the properties ofTheor, introducing the concept of diagonalizable algebra (see [7]): of course this algebraization can be applied to all these theories in which there exists a predicate with analogous properties. In this paper, by means of methods of universal algebra, we study the equational class of diagonalizable algebras, proving, among other things, that the set of identities satisfied byTheor which are consequences of the known ones is decidable.

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References

  1. C. Bernardi,The fixed-point theorem for diagonalizable algebras (The algebraization of the theories which express Theor; III),Studia Logica, Vol. 34, No. 3 (1975), pp. 239–251.

    Google Scholar 

  2. R. Franci,Idealità di alcune classi di algebre di Boole con operatori, to appear inBoll. Un. Mat. Ital.

  3. G. Grätzer,Universal Algebra, Van Nostrand, Princeton, 1968.

    Google Scholar 

  4. P. R. Halmos,Algebraic logic, I. Monadic Boolean algebras,Compositio Mathematicae, Vol. 12 (1955), pp. 217–249. (reprinted inAlgebraic logic, Chelsea Publishing Company, New York (1962)).

    Google Scholar 

  5. M. Löb,Solution of a problem of Leon Henkin,Journal of Symbolic Logic, Vol. 20 (1955), pp. 115–118.

    Google Scholar 

  6. R. Magari,Problemi aperti sulle algebre diagonali, to appear inRend. Sem. Mat. Fis. Milano.

  7. R. Magari,The diagonalizable algebras (the algebraization of the theories which express Theor; II) to appear inBoll. Un. Mat. Ital.

  8. R. Magari,Representation and duality theory for diagonalizable algebras (The algebraization of the theories which express Theor; IV), this volume.

  9. F. Montagna,For every n, the n-freely generaled algebra is not functionally free in the equational class of diagonalizable algebras (The algebraization of the theories which express Theor; V), this volume.

  10. G. Marongiu,Sequenze di predicati aritmetici di tipo «teor»,Boll. Un. Mat. Ital., IV, Vol. IX (1974), pp. 361–375.

    Google Scholar 

  11. E. Mendelson,Introduction to Mathematical Logic, Van Nostrand, Princeton (1964).

    Google Scholar 

  12. P. Pagli,On some extensions of Diagonalization lemma in Lindenbaum algebra of Peano arithmetic (the algebraization of theories which express Theor; I), to appear.

  13. G. Sambin,Un'estensione del teorema di Löb, to appear inRend. Sem. Mat. Univ. Padova.

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Authors and Affiliations

  1. Siena University, Siena, Italy

    Glaudio Bernardi

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  1. Glaudio Bernardi
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Allatum est die 21 Aprilis 1975

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Bernardi, G. On the equational class of diagonalizable algebras. Stud Logica 34, 321–331 (1975). https://doi.org/10.1007/BF02121663

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  • DOI: https://doi.org/10.1007/BF02121663

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