Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level
Institute for Information Transmission Problems of the Russian Academy of Sciences, 127051 Moscow, Russia
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Mathematics 2020, 8(6), 910; https://doi.org/10.3390/math8060910
Received: 11 May 2020 / Revised: 25 May 2020 / Accepted: 27 May 2020 / Published: 3 June 2020
(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)
Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that $n\ge 2$ . Then: 1. If it holds in the constructible universe $\mathbf{L}$ that $a\subseteq \omega $ and $a\notin {\Sigma}_{n}^{1}\cup {\Pi}_{n}^{1}$ , then there is a generic extension of $\mathbf{L}$ in which $a\in {\mathsf{\Delta}}_{n+1}^{1}$ but still $a\notin {\Sigma}_{n}^{1}\cup {\Pi}_{n}^{1}$ , and moreover, any set $x\subseteq \omega $ , $x\in {\Sigma}_{n}^{1}$ , is constructible and ${\Sigma}_{n}^{1}$ in $\mathbf{L}$ . 2. There exists a generic extension $\mathbf{L}$ in which it is true that there is a nonconstructible ${\mathsf{\Delta}}_{n+1}^{1}$ set $a\subseteq \omega $ , but all ${\Sigma}_{n}^{1}$ sets $x\subseteq \omega $ are constructible and even ${\Sigma}_{n}^{1}$ in $\mathbf{L}$ , and in addition, $\mathbf{V}=\mathbf{L}\left[a\right]$ in the extension. 3. There exists an generic extension of $\mathbf{L}$ in which there is a nonconstructible ${\Sigma}_{n+1}^{1}$ set $a\subseteq \omega $ , but all ${\mathsf{\Delta}}_{n+1}^{1}$ sets $x\subseteq \omega $ are constructible and ${\mathsf{\Delta}}_{n+1}^{1}$ in $\mathbf{L}$ . Thus, nonconstructible reals (here subsets of $\omega $ ) can first appear at a given lightface projective class strictly higher than ${\Sigma}_{2}^{1}$ , in an appropriate generic extension of $\mathbf{L}$ . The lower limit ${\Sigma}_{2}^{1}$ is motivated by the Shoenfield absoluteness theorem, which implies that all ${\Sigma}_{2}^{1}$ sets $a\subseteq \omega $ are constructible. Our methods are based on almostdisjoint forcing. We add a sufficient number of generic reals to $\mathbf{L}$ , which are very similar at a given projective level n but discernible at the next level $n+1$ .
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Keywords:
definability; nonconstructible reals; projective hierarchy; generic models; almost disjoint forcing
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Doi: 10.3390/math8060910
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Kanovei, V.; Lyubetsky, V. Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level. Mathematics 2020, 8, 910.
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