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Article

On the Δ n 1 Problem of Harvey Friedman

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
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Author to whom correspondence should be addressed.
Dedicated to the 70-th anniversary of A. L. Semenov.
These authors contributed equally to this work.
Mathematics 2020, 8(9), 1477; https://doi.org/10.3390/math8091477
Received: 2 August 2020 / Revised: 22 August 2020 / Accepted: 23 August 2020 / Published: 1 September 2020
(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)
In this paper, we prove the following. If n3, then there is a generic extension of L, the constructible universe, in which it is true that the set P(ω)L of all constructible reals (here—subsets of ω) is equal to the set P(ω)Δn1 of all (lightface) Δn1 reals. The result was announced long ago by Leo Harrington, but its proof has never been published. Our methods are based on almost-disjoint forcing. To obtain a generic extension as required, we make use of a forcing notion of the form Q=C×νQν in L, where C adds a generic collapse surjection b from ω onto P(ω)L, whereas each Qν, ν<ω2L, is an almost-disjoint forcing notion in the ω1-version, that adjoins a subset Sν of ω1L. The forcing notions involved are independent in the sense that no Qν-generic object can be added by the product of C and all Qξ, ξν. This allows the definition of each constructible real by a Σn1 formula in a suitably constructed subextension of the Q-generic extension. The subextension is generated by the surjection b, sets Sω·k+j with jb(k), and sets Sξ with ξω·ω. A special character of the construction of forcing notions Qν is L, which depends on a given n3, obscures things with definability in the subextension enough for vice versa any Δn1 real to be constructible; here the method of hidden invariance is applied. A discussion of possible further applications is added in the conclusive section. View Full-Text
Keywords: Harvey Friedman’s problem; definability; nonconstructible reals; projective hierarchy; generic models; almost-disjoint forcing Harvey Friedman’s problem; definability; nonconstructible reals; projective hierarchy; generic models; almost-disjoint forcing
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MDPI and ACS Style

Kanovei, V.; Lyubetsky, V. On the Δ n 1 Problem of Harvey Friedman. Mathematics 2020, 8, 1477. https://doi.org/10.3390/math8091477

AMA Style

Kanovei V, Lyubetsky V. On the Δ n 1 Problem of Harvey Friedman. Mathematics. 2020; 8(9):1477. https://doi.org/10.3390/math8091477

Chicago/Turabian Style

Kanovei, Vladimir; Lyubetsky, Vassily. 2020. "On the Δ n 1 Problem of Harvey Friedman" Mathematics 8, no. 9: 1477. https://doi.org/10.3390/math8091477

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