The Stable core

2012 ◽  
Vol 18 (2) ◽  
pp. 261-267 ◽  
Author(s):  
Sy-david Friedman
Keyword(s):  
Inner Model ◽  
Definable Sets ◽  
Stable Core ◽  
Entire Universe ◽  
Definition Of ◽  
Simple Definition

AbstractVopěnka proved long ago that every set of ordinals is set-generic over HOD, Gödel's inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over (HOD,S), and indeed over the even smaller inner model =(L[S],S), where S is the Stability predicate. We refer to the inner model as the Stable Core of V. The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core (this is not possible for HOD by Vopěnka's theorem).

A simple definition of ‘intentionally’

Cognition ◽  
2021 ◽  
Vol 214 ◽  
pp. 104806
Author(s):  
Tadeg Quillien ◽  
Tamsin C. German
Keyword(s):  
Definition Of ◽  
Simple Definition

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

scholarly journals Nærvær - forsøg på en nem definition af leg

2001 ◽  
Vol 17 ◽  
Author(s):  
Jørn Møller
Keyword(s):  
Phenomenological Anthropology ◽  
The Aesthetic ◽  
Definition Of ◽  
Simple Definition

En fænomenologisk og antropologisk analyse af legen som et universelt, eksistentielt rum for mennesket.Presence – An attempt at an easy definition of playIn this article a simple definition of play has been attempted within the framework of phenomenological anthropology. When the process is seen as more serious than the result, we play. When, on the contrary, the result is the more important, we work. To express it in simple formulas in which P is process and R, result:P/R > 1 = Play P/R < 1 = WorkThe definition is discussed in relation to a number of approsing rations which normally provoke reflexive efforts: play vs. seriousness; play vs. labour; play vs. reality; play vs. war; play vs. sport; play vs. ritual; play as symbol and metaphor.Further, the phenomenological concept of play is contrasted to the pedagogical concept, in an analysis of the instrumental use of play vs. play as a meaningful practise in itself.Finally, reference is made to W.H. Auden’s view of the aesthetic process as a dual act involving sin and the search for redemption.

scholarly journals Photojournalism as a Creator of Values: otherness in Sámi representations

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
Heli Lehtela
Keyword(s):  
Content Analysis ◽  
Ethnic Minority ◽  
Ethnic Minorities ◽  
The Arctic ◽  
Value Judgments ◽  
Definition Of ◽  
Simple Definition

The research draws on the insights provided by photojournalism to examine the values and value formation of journalism in relation to power and ethnic minorities. News photos are representations which affect emotions; they engender meanings about reality and settings more subtly than texts. Photojournalism makes value judgments on issues, groups and persons through the size of the photos, settings, poses, the roles depicted, and simply by choosing who is photographed. The material to be analysed consists of news photos dealing with the Sámi published in the Arctic press. My particular focus is the news items dealing with the Sámi and how the ethnic minority is depicted in those representations. The methods to be used are content analysis and formal photo analysis. I use the concept of otherness for determining journalists’ valuations. A simple definition of otherness could be ‘being an outsider’.

On the axiomatic foundations of the theory of Hermitian forms

1970 ◽  
Vol 67 (2) ◽  
pp. 243-250 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Recent Work ◽  
Quadratic Forms ◽  
Hermitian Form ◽  
Major Objective ◽  
Hermitian Forms ◽  
Division Rings ◽  
General Terms ◽  
Definition Of ◽  
Simple Definition

In recent work on some topological problems (7), I was forced to adopt a complicated definition of ‘Hermitian form’ which differed from any in the literature. A recent paper by Tits(5) on quadratic forms over division rings contains a new and simple definition of these. A major objective of this paper is to formulate both these definitions in somewhat more general terms, and to show that they are equivalent.

Extending Gödel's negative interpretation to ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 221-229 ◽  
Author(s):  
William C. Powell
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Stable Sets ◽  
Class Theory ◽  
Heyting Arithmetic ◽  
Inner Model ◽  
Order Arithmetic ◽  
Definition Of ◽  
Excluded Middle ◽  
Negative Sense

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

Sodic soils - New perspectives

Soil Research ◽  
1993 ◽  
Vol 31 (6) ◽  
pp. 683 ◽  
Author(s):  
ME Sumner
Keyword(s):  
Electrolyte Concentration ◽  
Exchangeable Cation ◽  
Sodic Soils ◽  
The World ◽  
Critical Flocculation Concentration ◽  
Low Levels ◽  
Definition Of ◽  
Physical Degradation ◽  
Soil Behaviour ◽  
Simple Definition

There are large areas of the world where soils are adversely affected by the presence of sodium (Na) as an exchangeable cation. Unlike their saline counterparts which are more extensive, sodic soils have received less attention in the literature. There has been considerable disagreement concerning the definition of sodicity, owing largely to the fact that many experiments used in the development of definitions did not account for the presence of salts in the water used to measure hydraulic properties. These problems are discussed and the conclusion is reached that a single simple definition is no longer possible. This problem is further exacerbated by the fact that many soils which would never have fallen into a previously defined sodic category, do in fact exhibit sodic properties. The major focus of this account of sodicity will therefore be the soils which contain relatively low levels of exchangeable Na. As such soils are widespread in both humid and subhumid areas of the world and are responsible for the production of a large proportion of the world cereal crop, they deserve special attention. Because swelling and dispersion are the primary processes responsible for the degradation of soil physical properties in the presence of Na, an account of clay behaviour in relation to Na and electrolyte concentration is presented before exploring these new realms of sodicity. Pure clay systems are not always suitable for use as models of soil behaviour in terms of dispersion and flocculation. However, as far as swelling is concerned, the correspondence is much better. Nevertheless, the effects of the exchangeable cations on dispersion are predictable albeit usually only qualitatively. This is partly due to the phenomenon of 'demixing' in which the cations are not distributed over all surfaces in the same proportions. The effects of Na and electrolyte concentration in relation to hydraulic conductivity, infiltration, crusting, runoff, erosion and hardsetting are discussed from which it emerges that the effects of Na are manifested in measurable and often sizeable proportions down to very low levels far below those previously used to define sodic soils. The primary processes responsible for physical degradation are swelling at relatively high levels and clay dispersion throughout the range of exchangeable Na percentage (ESP). Provided that the total electrolyte concentration (TEC) is below the critical flocculation concentration (CFC), clays will disperse spontaneously at high ESP values, whereas at lower ESP levels, inputs of energy are required for dispersion. The TEC of the ambient solution, because of its effects in promoting clay flocculation, is crucial in determining soil physical behaviour.

APPLICATION OF THE SZNAJD SOCIOPHYSICS MODEL TO SMALL-WORLD NETWORKS

2001 ◽  
Vol 12 (10) ◽  
pp. 1537-1544 ◽  
Author(s):  
A. S. ELGAZZAR
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Time Series Analysis ◽  
Fixed Points ◽  
Small World ◽  
Small World Networks ◽  
Opinion Formation ◽  
Series Analysis ◽  
Definition Of ◽  
Simple Definition

The Sznajd model for the opinion formation is generalized to small-world networks. This generalization destroyed the stalemate fixed point. Then a simple definition of leaders is included. No fixed points are observed. This model displays some interesting aspects in sociology. The model is investigated using time series analysis.

Working hard to make a simple definition of synergies

2016 ◽  
Vol 17 ◽  
pp. 24-26 ◽  
Author(s):  
Cristiano Alessandro ◽  
Filipe Oliveira Barroso ◽  
Matthew Tresch
Keyword(s):  
Definition Of ◽  
Simple Definition
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