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# 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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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 $n≥3$, 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 $Q=Cℂ×∏νQν$ 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 $Qξ$, $ξ≠ν$. 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 $j∈b(k)$, and sets $Sξ$ with $ξ≥ω·ω$. A special character of the construction of forcing notions $Qν$ is $L$, which depends on a given $n≥3$, 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
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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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