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 Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 3, Pages 550–579 (Mi izv1779)

On the nonemptiness of classes in axiomatic set theory

V. G. Kanovei

Abstract: Theorems are proved on the consistency with $ZF$, for $n\geqslant2$, of each of the following three propositions: (1) there exists an $L$-minimal (in particular, nonconstructive) $a\subseteq\omega$ such that $V=L[a]$ and $\{a\}\in\Pi_n^1$, but every $b\subseteq\omega$ of class $\Sigma_n^1$ with constructive code is itself constructive; (2) there exist $a,b\subseteq\omega$ such that their $L$-degrees differ by a formula from $\Pi_n^1$, but not by formulas from $\Sigma_n^1$ with constants from $L$ ($X$ and $Y$ are said to differ by a formula $\sim[(\exists x\in X)\varphi(x)\equiv(\exists y\in Y)\varphi(y)])$; (3) there exists an infinite, but Dedekind finite, set $X\in\mathscr P(\omega)$ of class $\Pi_n^1$, whereas there are no such sets of class $\underline\Sigma_n^1$. The proof uses Cohen's forcing method.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1978, 12:3, 507–535

Bibliographic databases:

UDC: 51.01.16
MSC: Primary 03E30; Secondary 03E35
Revised: 22.02.1977

Citation: V. G. Kanovei, “On the nonemptiness of classes in axiomatic set theory”, Izv. Akad. Nauk SSSR Ser. Mat., 42:3 (1978), 550–579; Math. USSR-Izv., 12:3 (1978), 507–535

Citation in format AMSBIB
\Bibitem{Kan78} \by V.~G.~Kanovei \paper On the nonemptiness of classes in axiomatic set theory \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1978 \vol 42 \issue 3 \pages 550--579 \mathnet{http://mi.mathnet.ru/izv1779} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=503431} \zmath{https://zbmath.org/?q=an:0427.03044|0409.03031} \transl \jour Math. USSR-Izv. \yr 1978 \vol 12 \issue 3 \pages 507--535 \crossref{https://doi.org/10.1070/IM1978v012n03ABEH001997} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. G. Kanovei, “The set of all analytically definable sets of natural numbers can be defined analytically”, Math. USSR-Izv., 15:3 (1980), 469–500
2. B. L. Budinas, “On the selector principle and analytic definability of constructive sets”, Russian Math. Surveys, 37:2 (1982), 207–208
3. Kanovei V. Lyubetsky V., “Definable Minimal Collapse Functions At Arbitrary Projective Levels”, J. Symb. Log., 84:1 (2019), 266–289
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