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On the ‘Definability of Definable’ Problem of Alfred Tarski

Author

Listed:
  • Vladimir Kanovei

    (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
    These authors contributed equally to this work.)

  • Vassily Lyubetsky

    (Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

In this paper we prove that for any m ≥ 1 there exists a generic extension of L , the constructible universe, in which it is true that the set of all constructible reals (here subsets of ω ) is equal to the set D 1 m of all reals definable by a parameter free type-theoretic formula with types bounded by m , and hence the Tarski ‘definability of definable’ sentence D 1 m ∈ D 2 m (even in the form D 1 m ∈ D 21 ) holds for this particular m . This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal.

Suggested Citation

  • Vladimir Kanovei & Vassily Lyubetsky, 2020. "On the ‘Definability of Definable’ Problem of Alfred Tarski," Mathematics, MDPI, vol. 8(12), pages 1-36, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2214-:d:461599
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