Ordinal definability in Jensen's model

1984 ◽  
Vol 49 (2) ◽  
pp. 608-620 ◽  
Author(s):  
Włodzimierz Zadrożny
Keyword(s):  
Boolean Algebras ◽  
Almost Disjoint ◽  
Partial Orderings ◽  
Disjoint Sets

In [J] R. Jensen proved that any model of ZFC can be generically extended to a model of ZFC + (∃a ⊆ ω)(V = L[a]). This extension was made via a class P of forcing conditions. One can ask what can be said about the class HOD in such extensions. In particular one can ask whether V = HOD holds in Jensen's model. A full answer to this question is given by the following assertion:Theorem. Let N = L[a] be given by forcing with Jensen's conditions (the classP).Then:1. N ⊨ V ≠ HOD,2. N ⊨ (∀n < ω)(HODn + 1 ⊈ HODω).3. N ⊨ (∀α ≥ ω)(HODα = HODω), or equivalently HODω + 1 = HODω.The result is established by investigating homogeneity properties of certain complete Boolean algebras that are associated with the partial orderings of forcing conditions for coding using almost disjoint sets.We explain shortly the first of our two technical results: It is rather well known that if we have a subset then by performing threefold almost disjoint coding we come to a set a0 ⊆ ω such that ; cf. [JS]. This is achieved by iterating over almost disjoint forcing . That means we first code by a subset , of [ω1ω2) using the almost disjoint forcing conditions . Then we code by aω, contained in [ω, ω1), using forcing with over .

Families of Almost Disjoint Sets

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

Strongs-Sequences and Variations on Martin's Axiom

1985 ◽  
Vol 37 (4) ◽  
pp. 730-746 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Pairwise Disjoint ◽  
Disjoint Family ◽  
Uncountable Family ◽  
Martin’S Axiom ◽  
Finite Set ◽  
Almost Disjoint ◽  
Disjoint Sets ◽  
Almost Disjoint Family ◽  
Require Equality ◽  
Single Set

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

Recursion theory on orderings. I. A model theoretic setting

1979 ◽  
Vol 44 (3) ◽  
pp. 383-402 ◽  
Author(s):  
G. Metakides ◽  
J.B. Remmel
Keyword(s):  
Algebraic Closure ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Formal Definition ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Partial Orderings ◽  
Definition Of ◽  
Algebraically Closed Fields

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a setS, denoted cl(S), is given in §1, however in vector spaces, cl(S) is just the subspace generated byS, in Boolean algebras, cl(S) is just the subalgebra generated byS, and in algebraically closed fields, cl(S) is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl(S) = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings.

Almost disjoint sets and Martin's axiom

1979 ◽  
Vol 44 (3) ◽  
pp. 313-318 ◽  
Author(s):  
Michael L. Wage
Keyword(s):  
Partial Order ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Almost Disjoint ◽  
Disjoint Sets

AbstractWe present a number of results involving almost disjoint sets and Martin's axiom. Included is an example, due to K. Kunen, of a c.c.c. partial order without property K whose product with every c.c.c. partial order is c.c.c.

Elementary embedding between countable Boolean algebras

1991 ◽  
Vol 56 (4) ◽  
pp. 1212-1229
Author(s):  
Robert Bonnet ◽  
Matatyahu Rubin
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Complete Theory ◽  
Linear Ordering ◽  
Elementary Embedding ◽  
Linear Orderings ◽  
Partial Orderings ◽  
Atomless Boolean Algebra ◽  
Atomic Boolean Algebra ◽  
Quasi Ordering

AbstractFor a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B2 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then ‹MT, ≤› is well-quasi-ordered. ∎ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ s is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that ‹A, < › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {ai, ⃒ i ∈ ω} ⊆ A, there are i < j < ω such that ai ≤ aj. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and are isomorphic. ∎ Let M′0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M′0, let B1 ≤′ B2 mean that B1 is embeddable in B2. Remark. ‹M′0, ≤′› is well-quasi-ordered. ∎ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.

scholarly journals Ultrafilters and almost disjoint sets

1974 ◽  
Vol 4 (3) ◽  
pp. 269-282 ◽  
Author(s):  
Karel Prikry
Keyword(s):  
Almost Disjoint ◽  
Disjoint Sets

scholarly journals Strongly almost disjoint sets and weakly uniform bases

2000 ◽  
Vol 352 (11) ◽  
pp. 4971-4987 ◽  
Author(s):  
Z. T. Balogh ◽  
S. W. Davis ◽  
W. Just ◽  
S. Shelah ◽  
P. J. Szeptycki
Keyword(s):  
Almost Disjoint ◽  
Disjoint Sets

Continuum cardinals generalized to Boolean algebras

2001 ◽  
Vol 66 (4) ◽  
pp. 1928-1958 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Boolean Algebras ◽  
Infinite Cardinal ◽  
Cardinal Numbers ◽  
Almost Disjoint ◽  
Infinite Sets ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.

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