On the ‘Definability of Definable’ Problem of Alfred Tarski

Vladimir Kanovei and Vassily Lyubetsky
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Vladimir Kanovei: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
Vassily Lyubetsky: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia

Mathematics, 2020, vol. 8, issue 12, 1-36

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 ? ) 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.

Keywords: definability of definable; tarski problem; type theoretic hierarchy; generic models; almost disjoint forcing (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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