Subdifferentials in Boolean-valued models of set theory

1984 ◽  
Vol 24 (5) ◽  
pp. 735-746 ◽  
Author(s):  
A. G. Kusraev ◽  
S. S. Kutateladze
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

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

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

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

Elementary extensions of models of set theory

2000 ◽  
Vol 39 (7) ◽  
pp. 509-514 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

scholarly journals Chains of end elementary extensions of models of set theory

1998 ◽  
Vol 63 (3) ◽  
pp. 1116-1136 ◽  
Author(s):  
Andrés Villaveces
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Elementary Extension ◽  
Models Of Set Theory

AbstractLarge cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (‘unfoldable cardinals’) lie in the boundary of the propositions consistent with ‘V = L’ and the existence of 0#. We also provide an ‘embedding characterisation’ of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

scholarly journals Almost-Free E-Rings of Cardinality ℵ1

2003 ◽  
Vol 55 (4) ◽  
pp. 750-765
Author(s):  
Rüdiger Göbel ◽  
Saharon Shelah ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Set Theory ◽  
Regular Cardinal ◽  
Additive Structure ◽  
Unital Ring ◽  
Ring Element ◽  
Models Of Set Theory

AbstractAn E-ring is a unital ring R such that every endomorphism of the underlying abelian group R+ is multiplication by some ring element. The existence of almost-free E-rings of cardinality greater than 2ℵ0 is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ℵ1 ≤ λ 2ℵ0 we construct E-rings of cardinality λ in ZFC which have ℵ1-free additive structure. For λ = ℵ1 we therefore obtain the existence of almost-free E-rings of cardinality ℵ1 in ZFC.

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