Omitting types and the real line

1987 ◽  
Vol 52 (4) ◽  
pp. 1020-1026 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Real Line ◽  
Logical Structure ◽  
Point Of View ◽  
Ground Model ◽  
The Real ◽  
The Family ◽  
Omitting Types ◽  
Countable Theory ◽  
Meager Sets ◽  
The Ideal

We investigate some relations between omitting types of a countable theory and some notions defined in terms of the real line, such as for example the ideal of meager subsets ofR. We also try to express connections between the logical structure of a theory and the existence of its countable models omitting certain families of types.It is well known that assuming MA we can omit <nonisolated types. But MA is rather a strong axiom. We prove that in order to be able to omit <nonisolated types it is sufficient to assume that the real line cannot be covered by less thanmeager sets; and this is in fact the weakest possible condition. It is worth pointing out that by means of forcing we can easily obtain the model of ZFC in whichRcannot be covered by <meager sets. It suffices to add to the ground modelCohen generic reals.We also formulate similar results for omitting pairwise contradictory types. It turns out that from some point of view it is much more difficult to find the family of pairwise contradictory types which cannot be omitted by a model ofT, than to find such a family of possibly noncontradictory types. Moreover, for any two countable theoriesT1,T2without prime models, the existence of a family ofκtypes which cannot be omitted by a model ofT1is equivalent to the existence of such a family forT2. This means that from the point of view of omitting types all theories without prime models are identical. Similar results hold for omitting pairwise contradictory types.

Omitting types for stable ccc theories

1990 ◽  
Vol 55 (3) ◽  
pp. 1037-1047 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Real Line ◽  
Basic Knowledge ◽  
Complete Theory ◽  
The Real ◽  
Special Cases ◽  
Standard Set ◽  
Tarski Algebra ◽  
Independence Results ◽  
Omitting Types ◽  
Meager Sets

In this paper we investigate omitting types for a certain kind of stable theories which we call stable ccc theories. In Theorem 2.1 we improve Steinhorn's result from [St]. We prove also some independence results concerning omitting types. The main results presented in this paper were part of the author's Ph.D. thesis [N1].Throughout, we use the standard set-theoretic and model-theoretic notation, such as can be found for example in [Sh] or [M]. So in particular T is always a countable complete theory in the language L. We consider all models of T and all sets of parameters subsets of the monster model ℭ, which is very saturated. Ln(A) denotes the Lindenbaum-Tarski algebra of formulas with parameters from A and n free variables. We omit n in Ln(A) when n = 1 or when it is clear from the context what n is. If φ, ψ ∈ L(A) are consistent then we say that φ is below ψ if ψ⊢ψ. For a type p and a set A ⊆ ℭ, p(A) is the set of tuples of elements of A which satisfy p. Formulas are special cases of types. We say that a type p is isolated over A if, for some φ() ∈ L(A), φ() ⊢ p(x), i.e. φ isolates p. For a formula φ, [φ] denotes the class of types which contain φ. We assume that the reader is familiar with some basic knowledge of forking, as presented in [Sh, III] or [M].Throughout, we work in ZFC. and denote (countable) transitive models of ZFC. cov K is the minimal number of meager sets covering the real line R. In this paper we prove theorems showing connections between omitting types and the combinatorics of the real line. More results in this direction are presented in [N2] and [N3].

Lebesque measure zero subsets of the real line and an infinite game

1996 ◽  
Vol 61 (1) ◽  
pp. 246-249 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
Measure Zero ◽  
Cardinal Number ◽  
Minimal Cardinality ◽  
Combinatorial Characterization ◽  
The Real ◽  
Lebesque Measure ◽  
The Ideal

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

Virtual and Ideal Readers of Browning's “Pan and Luna”: the Drama in the Dramatic Idyl

1987 ◽  
Vol 15 ◽  
pp. 151-160 ◽  
Author(s):  
Betty S. Flowers
Keyword(s):  
Point Of View ◽  
Reading Public ◽  
Internal Auditor ◽  
The Real ◽  
The Ideal

Most of “Pan and Luna” is addressed not to an internal auditor but to what Gerald Prince calls the “virtual reader,” the reader the author imagines himself or herself to be writing to – in the case of “Pan and Luna,” the Victorian reading public. Prince observes:Every author, provided he is writing for someone other than himself, develops his narrative as a function of a certain type of reader whom he bestows with certain qualities, faculties, and inclinations according to his opinion of men in general (or in particular) and according to the obligations he feels should be respected. This virtual reader is different from the real reader: writers frequently have a public they don't deserve. (9)In addition to the distinction between the virtual reader and the real reader, Prince makes a further distinction between the real reader and the ideal reader. From the writer's point of view, “an ideal reader would be one who would understand perfectly and would approve entirely the least of his words, the most subtle of his intentions” (9).

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

𝜎-ideals and outer measures on the real line

2019 ◽  
Vol 25 (1) ◽  
pp. 25-36
Author(s):  
Salvador Garcia-Ferreira ◽  
Artur H. Tomita ◽  
Yasser Ferman Ortiz-Castillo
Keyword(s):  
Real Line ◽  
Weak Selection ◽  
The Real ◽  
The Family

Abstract A weak selection on {\mathbb{R}} is a function {f\colon[\mathbb{R}]^{2}\to\mathbb{R}} such that {f(\{x,y\})\in\{x,y\}} for each {\{x,y\}\in[\mathbb{R}]^{2}} . In this article, we continue with the study (which was initiated in [1]) of the outer measures {\lambda_{f}} on the real line {\mathbb{R}} defined by weak selections f. One of the main results is to show that CH is equivalent to the existence of a weak selection f for which {\lambda_{f}(A)=0} whenever {\lvert A\rvert\leq\omega} and {\lambda_{f}(A)=\infty} otherwise. Some conditions are given for a σ-ideal of {\mathbb{R}} in order to be exactly the family {\mathcal{N}_{f}} of {\lambda_{f}} -null subsets for some weak selection f. It is shown that there are {2^{\mathfrak{c}}} pairwise distinct ideals on {\mathbb{R}} of the form {\mathcal{N}_{f}} , where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that {\mathcal{N}_{f}} is exactly the σ-ideal of meager subsets of {\mathbb{R}} . Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of {\lambda_{f}} -measurable sets.

Construction Scope: Quality, Budget and Timeline ‘Small’ Endeavours, Portuguese Context

2017 ◽  
Vol 864 ◽  
pp. 369-377
Author(s):  
Nuno Dinis Costa Areias Cortiços
Keyword(s):  
Real Estate ◽  
Production Process ◽  
Active Role ◽  
Point Of View ◽  
Construction Sector ◽  
Good Practices ◽  
The Real ◽  
Real Estate Developer ◽  
Construction Services ◽  
The Ideal

The trinomial of the scope, quality, budget and timeline, is an objective by excellence, pursued by those with an active role in the production process, to which the construction sector is no stranger. The ideal is to ensure the satisfaction of the three parameters/variables, by those who seek construction services: Project Owner. Why the Scope is not reached?! The causes are divided in two levels: insufficient training, rooted in the Sector, that favours informality; and, lack of data to understand the causes, resulting in no definition or application of corrective measures (good practices guides). The focus of this article is the 'small endeavours', managed by: (1) Real Estate Developer, in control of the different phases, from municipal licensing, through the project and construction license, the remaining will be undertaken by a Builder or Investor, for subsequent sale; or (future) Owner without control of the initial decisions; the concerns of the latter are bound by the binomial Timeline and Budget; or, (2) P.O., from the outset, take control of all phases, always faithful to his idea of the Scope; introduces adjustments and changes defending his point of view or interest without the perception of the real impact of those on the overall satisfaction, focused on the fulfilment of quality achieved.

The Baire category theorem and cardinals of countable cofinality

1982 ◽  
Vol 47 (2) ◽  
pp. 275-288 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Real Line ◽  
Measure Zero ◽  
Baire Category ◽  
The Other ◽  
Baire Category Theorem ◽  
The Real ◽  
Zero Sets ◽  
Category Theorem ◽  
Dense Sets ◽  
The Ideal

AbstractLet κB be the least cardinal for which the Baire category theorem fails for the real line R. Thus κB is the least κ such that the real line can be covered by κ many nowhere dense sets. It is shown that κB cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for 2ω1 be ℵω. Similar questions are considered for the ideal of measure zero sets, other ω1, saturated ideals, and the ideal of zero-dimensional subsets of Rω1.

On the generalization of density topologies on the real line

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Jacek Hejduk ◽  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Density Theorem ◽  
Density Topology ◽  
The Real ◽  
Lebesgue Density ◽  
The Family ◽  
Real Anal ◽  
Similarities And Differences ◽  
Relevant Operator

AbstractThe paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

scholarly journals The Meaning of Time and Covariant Superderivatives in Supermechanics

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Gil Salgado ◽  
José A. Vallejo-Rodríguez
Keyword(s):  
Real Line ◽  
Point Of View ◽  
The Real ◽  
Theoretic Point ◽  
Group Law

We present a review of the basics of supermanifold theory (in the sense of Berezin-Kostant-Leites-Manin) from a physicist's point of view. By considering a detailed example of what does it mean the expression “to integrate an ordinary superdifferential equation” we show how the appearance of anticommuting parameters playing the role of time is very natural in this context. We conclude that in dynamical theories formulated whithin the category of supermanifolds, the space that classically parametrizes time (the real lineℝ) must be replaced by the simplest linear supermanifoldℝ1|1. This supermanifold admits several different Lie supergroup structures, and we analyze from a group-theoretic point of view what is the meaning of the usual covariant superderivatives, relating them to a change in the underlying group law. This result is extended to the case ofN-supersymmetry.

On the closure of a class of subsets of the real line

1978 ◽  
Vol 83 (2) ◽  
pp. 181-182 ◽  
Author(s):  
I. Calvert
Keyword(s):  
Real Line ◽  
The Real ◽  
The Family

A subset, X, of R belongs to the family T(a) if |X| ≥ 2 and, for all x and y belonging to X, ax + (1 − a) y ∈ X, where a ∈ R.I consider the problem of determining for which values of a > 1 all elements of T(a) are dense in R.

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