Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers

1979 ◽  
Vol 18 (214) ◽  
pp. 0-0 ◽  
Author(s):  
Thomas Jech ◽  
Karel Prikry
Keyword(s):  
Almost Disjoint ◽  
Generic Ultrapowers

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 .

Families of Almost Disjoint Sets

1974 ◽  
pp. 286-310
Author(s):  
W. Wistar Comfort ◽  
Stylianos Negrepontis
Keyword(s):  
Almost Disjoint ◽  
Disjoint Sets

ALMOST DISJOINT FAMILIES OF COUNTABLE SETS

1984 ◽  
pp. 59-88 ◽  
Author(s):  
B. BALCAR ◽  
J. DOČKÁLKOVÁ ◽  
P. SIMON
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families

scholarly journals The Ramsey property implies no mad families

2019 ◽  
Vol 116 (38) ◽  
pp. 18883-18887 ◽  
Author(s):  
David Schrittesser ◽  
Asger Törnquist
Keyword(s):  
Set Theory ◽  
Large Set ◽  
Descriptive Set Theory ◽  
Ramsey Property ◽  
Mad Families ◽  
Almost Disjoint ◽  
Descriptive Set ◽  
Maximal Almost Disjoint ◽  
Borel Ideals ◽  
Standing Problem

We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

Comparing almost-disjoint families

1986 ◽  
Vol 47 (3-4) ◽  
pp. 321-323 ◽  
Author(s):  
P. Komjáth
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families

Fréchet-like properties and almost disjoint families

2020 ◽  
Vol 277 ◽  
pp. 107216
Author(s):  
César Corral ◽  
Michael Hrušák
Keyword(s):  
Almost Disjoint ◽  
Almost Disjoint Families
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