A COMBINATORIAL CORE OF THE ITERATED PERFECT SET MODEL

1999 ◽  
Vol 25 (1) ◽  
pp. 137
Author(s):  
Ciesielski
Keyword(s):  
Perfect Set

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

scholarly journals Review of Locke’s Metaphysics by Matthew Stuart

Locke Studies ◽  
2014 ◽  
Vol 14 ◽  
pp. 263-271
Author(s):  
Victor Nuovo
Keyword(s):  
Perfect Set ◽  
Navigational Aids

This is a substantial book in several ways. To begin with, it is great in bulk and long in content. But it is so well ordered that in spite of its length it can be navigated with ease. Its contents are topically arranged. Its seventy sections numbered consecutively are distributed among ten chapters. Its topics and subtopics are described in a table of contents and elaborated in summaries at the head of each chapter, which provide the reader with a running argument. With these aids, it is possible to survey the entire contents of the book in short order and to jump from one place to another without losing one’s way. A table of contents and index locorum complete a perfect set of navigational aids. The method is reminiscent of Locke.

Small Coverings with Smooth Functions under the Covering Property Axiom

2005 ◽  
Vol 57 (3) ◽  
pp. 471-493 ◽  
Author(s):  
Krzysztof Ciesielski ◽  
Janusz Pawlikowski
Keyword(s):  
Borel Function ◽  
Covering Property ◽  
Smooth Functions ◽  
Differentiable Functions ◽  
Perfect Set ◽  
Covering Property Axiom

AbstractIn the paper we formulate a Covering Property Axiom, CPAprism, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.(a) There exists a family ℱ of less than continuummany functions from ℝ to ℝ such that ℝ2 is covered by functions from ℱ, in the sense that for every 〈x, y〉 ∈ ℝ2 there exists an f ∈ ℱ such that either f (x) = y or f (y) = x.(b) For every Borel function f : ℝ → ℝ there exists a family ℱ of less than continuum many “” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of f.(c) For every n > 0 and a Dn function f: ℝ → ℝ there exists a family ℱ of less than continuum many Cn functions whose graphs cover the graph of f.We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiĭ.

scholarly journals Introducing Value Axiom for New Design Creations

2019 ◽  
Vol 301 ◽  
pp. 00002
Author(s):  
Masayuki Nakao ◽  
Kenji Iino
Keyword(s):  
Early Stage ◽  
Design Parameters ◽  
Independence Axiom ◽  
Design Equation ◽  
Functional Requirements ◽  
Monetary Value ◽  
Perfect Set ◽  
Information Axiom ◽  
Japanese Yen

This paper proposes “Value Axiom” that states “The larger the sum of Customer Attribute values, the better the design.” A customer evaluates a design with the sum of the value produced by each Customer Attribute, expressing it with a monetary value such as Japanese yen. A designer can hardly estimate and express a perfect set of Customer Attributes at the early stage of a design. The designer writes down the design equation to visualize the entire design, and improves the sets of Design Parameters and Functional Requirements using the Independence Axiom and Information Axiom, and at the same time, it is also important to review the values of Customer Attributes using the Value Axiom.

scholarly journals An application of a theorem of Singer

1974 ◽  
Vol 19 (2) ◽  
pp. 119-123 ◽  
Author(s):  
D. M. Connolly ◽  
J. H. Williamson
Keyword(s):  
Measure Zero ◽  
General Context ◽  
Measure Algebra ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Perfect Set ◽  
Convolution Power ◽  
Recent Theorem ◽  
Vector Sum ◽  
Positive Results

The authors have recently treated (2) the problem of finding subsets E of the real line , of type Fσ, such that E–E contains an interval and the k-fold vector sum (k)E is of measure zero. Positive results can be obtained, for all k, on the basis of a recent theorem of J. A. Haight (3), following earlier partial results (1), (4) for k ≦ 7; and indeed in these cases the problem has a solution with E a perfect set. An analogous problem, apparently in most respects subtler than the first, is the following. Do there exist finite regular Borel measures μ on such that is absolutely continuous (where is the adjoint of μ) and the kth convolution power μk is singular? Both problems are of interest in the general context of elucidating the properties of the measure algebra or, more generally, M(G) for locally compact abelian G. The second problem may be regarded as an attempt to provide (at least one aspect of) a multiplicity theory for the first.

scholarly journals Small sets containing any pattern

2018 ◽  
Vol 168 (1) ◽  
pp. 57-73
Author(s):  
URSULA MOLTER ◽  
ALEXIA YAVICOLI
Keyword(s):  
Hausdorff Measure ◽  
Analogous Result ◽  
Measure Zero ◽  
General Construction ◽  
Dimension Function ◽  
Perfect Set ◽  
Finite Set ◽  
Isolated Points ◽  
Special Case

AbstractGiven any dimension function h, we construct a perfect set E ⊆ ${\mathbb{R}}$ of zero h-Hausdorff measure, that contains any finite polynomial pattern.This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set E in ${\mathbb{R}}^N$, of h-Hausdorff measure zero, such that for any finite set {f1,. . .,fn} ⊆ $\mathcal{F}$, E satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$.We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points.

scholarly journals A Basis Theorem for Perfect Sets

1998 ◽  
Vol 4 (2) ◽  
pp. 204-209 ◽  
Author(s):  
Marcia J. Groszek ◽  
Theodore A. Slaman
Keyword(s):  
Set Theory ◽  
General Theorem ◽  
Sufficient Condition ◽  
Specific Instance ◽  
Basis Theorem ◽  
Perfect Set ◽  
Models Of Set Theory

AbstractWe show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair M ⊂ N of models of set theory implying that every perfect set in N has an element in N which is not in M.

scholarly journals Iterated perfect-set forcing

1979 ◽  
Vol 17 (3) ◽  
pp. 271-288 ◽  
Author(s):  
James E. Baumgartner ◽  
Richard Laver
Keyword(s):  
Perfect Set
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