Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

• Main Authors: Vladimir Kanovei, Vassily Lyubetsky
• Format: Article
• Language: English
• Published: MDPI AG 2020-06-01
• Series: Mathematics
• Subjects: definability; nonconstructible reals; projective hierarchy; generic models; almost disjoint forcing
• Online Access: https://www.mdpi.com/2227-7390/8/6/910

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. Then: 1. If it holds in the constructible universe <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula> that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>∉</mo> <msubsup> <mi>Σ</mi> <mi>n</mi> <mn>1</mn> </msubsup> <mo>∪</mo> <msubsup> <mi>Π</mi> <mi>n</mi> <mn>1</mn> </msubsup> </mrow> </semantics> </math> </inline-formula>, then there is a generic extension of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula> in which <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>∈</mo> <msubsup> <mi>Δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </mrow> </semantics> </math> </inline-formula> but still <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>∉</mo> <msubsup> <mi>Σ</mi> <mi>n</mi> <mn>1</mn> </msubsup> <mo>∪</mo> <msubsup> <mi>Π</mi> <mi>n</mi> <mn>1</mn> </msubsup> </mrow> </semantics> </math> </inline-formula>, and moreover, any set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>∈</mo> <msubsup> <mi>Σ</mi> <mi>n</mi> <mn>1</mn> </msubsup> </mrow> </semantics> </math> </inline-formula>, is constructible and <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mi>n</mi> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula>. 2. There exists a generic extension <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula> in which it is true that there is a nonconstructible <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula>, but all <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mi>n</mi> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> sets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula> are constructible and even <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mi>n</mi> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula>, and in addition, <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold">V</mi> <mo>=</mo> <mi mathvariant="bold">L</mi> <mo>[</mo> <mi>a</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> in the extension. 3. There exists an generic extension of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula> in which there is a nonconstructible <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula>, but all <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> sets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula> are constructible and <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula>. Thus, nonconstructible reals (here subsets of <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula>) can first appear at a given lightface projective class strictly higher than <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mn>2</mn> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula>, in an appropriate generic extension of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula>. The lower limit <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mn>2</mn> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> is motivated by the Shoenfield absoluteness theorem, which implies that all <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>Σ</mi> <mn>2</mn> <mn>1</mn> </msubsup> </semantics> </math> </inline-formula> sets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>⊆</mo> <mi>ω</mi> </mrow> </semantics> </math> </inline-formula> are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">L</mi> </semantics> </math> </inline-formula>, which are very similar at a given projective level <i>n</i> but discernible at the next level <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>.