Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ, κ + is greater than κ + of HOD. The proof uses a very general lemma showing that homogeneity is preserved through certain reverse Easton iterations.

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

I give reasons to suggest that the various ‘homogeneous substance’ objections to perdurance theory should not be regarded as raising serious difficulties. The main strategy is to show that there are equally exotic possibilities involving extended mereological simples that may turn the tables on the endurance theorist, insofar as she will have difficulties with these cases analogous to those she raises for the perdurantist. I conclude that such exotic cases are less useful that we might suppose in adjudicating between these (...) competing doctrines of persistence. (shrink)

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set C a sum of a countable number of pairwise disjoint h-homogeneous subspaces that are closed in X? It follows that every Borel set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \subset {\bf R}^n}$$\end{document} can be partitioned into countably many h-homogeneous subspaces that are Gδ-sets in X.

We prove that any torsion-free, residually finite relatively free group of infinite rank is not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} -homogeneous. This generalizes Sklinos’ result that a free group of infinite rank is not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} -homogeneous, and, in particular, gives a new simple proof of that result.

Wesley Salmon has recently proposed a new theory of scientific explanation based on a model which he calls the statistical-relevance model. It is intended primarily as an account of the structure of explanations of particular events--explanations which, according to Salmon, are very often motivated largely by practical concerns. Two important features of this account are the concepts of homogeneity and screening off. In this paper we argue that the employment of these two concepts (which, in fact, are intimately connected) (...) is fundamentally at odds with the desire to capture everyday explanations of the practical sort, and that it leads instead to an account in which the only satisfactory explanations are those expressed in the very deepest physical terms. (shrink)

The homogeneous gravitational field is obtained from a Schwarzschild field in the limit of infinite mass.

Assuming Jensen's principle ◊+ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todorčević.

My article aims to develop a relational, pluralistic political theory that moves beyond standard theories of liberal democracy, and to consider how such a theory translates into our public school settings. I use a narrative style argument to share stories that focus on homogeneity and diversity from my visit to a Japanese elementary school, as I consider, drawing on the work of Chantal Mouffe, the important role harmony and disagreement, and a tension between homogeneity and diversity, play in (...) encouraging citizens to contribute to their school and their larger communities in a democracy-always-in-the-making. I argue that there is much we can learn from Japanese educational practices. (shrink)

A Dedekind algebra is an order pair (B, h) where B is a non-empty set and h is a similarity transformation on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are 0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type which occur in the decomposition of (...) the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. It is shown that configuration signatures can be used to characterize the homogeneous, universal and homogeneous-universal Dedekind algebras. This characterization is used to prove various results about these subclasses of Dedekind algebras. (shrink)

This paper studies homogeneously Suslin (hom) sets of reals in tame mice. The following results are established: In 0 ¶ the hom sets are precisely the [Symbol] sets. In M n every hom set is correctly [Symbol] and (δ + 1)-universally Baire where ä is the least Woodin. In M u every hom set is <λ-hom, where λ is the supremum of the Woodins.

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0long does not exist, or else (b) V = K, where K is the core model below a μ-measurable cardinal.

The free Schrödinger equation is shown to be a consequence of spacetime homogeneity in the non-relativistic domain. This may help understand the origin of the wave equations in quantum theory.

We study non-homogeneity of quotients of Prikry and tree Prikry forcings with non-normal ultrafilters over some natural distributive forcing notions.

We consider the question of when, given a subset A of M, the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym(A). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω-stable structures s-homogeneity is preserved under naming countably many constants, (...) but under slightly weaker conditions it can be lost by naming a single point. (shrink)

We show that if M is a strongly minimal large homogeneous structure in a countable similarity type and the pregeometry of M is locally modular but not modular, then the pregeometry is affine over a division ring.

The Condorcet jury theorem (CJT) is based on the assumption of homogeneous voters who imperfectly know the correct policy. We reassess the validity of the CJT when voters are homogeneous and each knows the correct decision with an average probability of more than a half.

One determining characteristic of contemporary sociopolitical systems is their power over increasingly large and diverse populations. This raises questions about power relations between heterogeneous individuals and increasingly dominant and homogenizing system objectives. This article crosses epistemic boundaries by integrating computer engineering and a historicalphilosophical approach making the general organization of individuals within large-scale systems and corresponding individual homogenization intelligible. From a versatile archeological-genealogical perspective, an analysis of computer and social architectures is conducted that reinterprets Foucault’s disciplines and political anatomy to (...) establish the notion of politics for a purely technical system. This permits an understanding of system organization as modern technology with application to technical and social systems alike. Connecting to Heidegger’s notions of the enframing (Gestell) and a more primal truth (anfänglicheren Wahrheit), the recognition of politics in differently developing systems then challenges the immutability of contemporary organization. Following this critique of modernity and within the conceptualization of system organization, Derrida’s democracy to come (à venir) is then reformulated more abstractly as organizations to come. Through the integration of the discussed concepts, the framework of Large-Scale Systems Composed of Homogeneous Individuals (LSSCHI) is proposed, problematizing the relationships between individuals, structure, activity, and power within large-scale systems. The LSSCHI framework highlights the conflict of homogenizing system-level objectives and individual heterogeneity, and outlines power relations and mechanisms of control shared across different social and technical systems. (shrink)

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a (...) single CD theory T such that every homogeneous model of T has a PA degree. (shrink)

In this dissertation, I investigate the role of Aristotle's theory of science, and specifically the homogeneity criterion, for the set-up of Aristotle's metaphysics project as sketched in Met. IV, 1--3 and VI, 1. I argue that Aristotelian metaphysics takes shape through a dialectical argument stretching from Met. I to VI, 1. Along this path, both Aristotle's theory of science , and the Platonist competitor in metaphysics are constantly kept in view. They propel the argument at crucial stages. The resulting (...) metaphysics is compatible with Aristotle's theory of science, though it is not a direct application of it. Rather, metaphysics is a content-neutral complement to the content-specific sciences. The Platonist competitor remains refuted. Some of its central ideas and tools are nevertheless assimilated into Aristotelian metaphysics. ;Aristotle's postulate of a universal science of being seems to contradict a central requirement for a science: the homogeneity principle. According to the Posterior Analytics, every science has its specific subject matter , which it explores in terms of predicative content, and to which it is restricted. Being, does not constitute such a genos. Aristotle has to find a feature which can play a role for 'being' that is analogous to 'genos' in a special science. Aristotle finds it in the pros hen notion . Pros hen makes the content-neutral, focal structure of being transparent. It is not an extension of synonymy . With the help of pros hen, Aristotle can construct a content-neutral universal metaphysics project, solving the puzzles which arose in III, 1/2 precisely as a conflict between the questions which metaphysics should address and the homogeneity criterion for content-specific sciences. His theory of science remains fully intact for the sciences for which it was designed. The objections against Platonist metaphysics remain in place as well. The content-neutral analysis of being meets the expectations formulated for metaphysics in I, 1/2, on the basis of endoxa. Aristotle's project is constituted by what is commonly called 'general metaphysics'. It includes a consideration of the divine. (shrink)

In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in Hyttinen 167–182). We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see Hyttinen.

It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given (...) in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called “compatible” if they satisfy the Morette–Van Hove formula and “anti-compatible” if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged. (shrink)

We classify countable metrically homogeneous graphs of diameter 3.

In clinical studies, it is important to investigate the effectiveness of different therapeutic designs, especially, multiple treatment groups to one control group. The paper mainly studies homogeneity test of many-to-one risk differences from correlated binary data under optimal algorithms. Under Donner’s model, several algorithms are compared in order to obtain global and constrained MLEs in terms of accuracy and efficiency. Further, likelihood ratio, score, and Wald-type statistics are proposed to test whether many-to-one risk differences are equal based on optimal (...) algorithms. Monte Carlo simulations show the performance of these algorithms through the total averaged estimation error, SD, MSE, and convergence rate. Score statistic is more robust and has satisfactory power. Two real examples are given to illustrate our proposed methods. (shrink)

A partially ordered set is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of . Assuming the set-theoretic assumption ⋄, it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not -homogeneous. This is impossible in the countable case for k ≥ 4.

This paper develops a semantic solution to the puzzle of Free Choice permission. The paper begins with a battery of impossibility results showing that Free Choice is in tension with a variety of classical principles, including Disjunction Introduction and the Law of Excluded Middle. Most interestingly, Free Choice appears incompatible with a principle concerning the behavior of Free Choice under negation, Double Prohibition, which says that Mary can’t have soup or salad implies Mary can’t have soup and Mary can’t have (...) salad. Alonso-Ovalle 2006 and others have appealed to Double Prohibition to motivate pragmatic accounts of Free Choice. Aher 2012, Aloni 2018, and others have developed semantic accounts of Free Choice that also explain Double Prohibition. -/- This paper offers a new semantic analysis of Free Choice designed to handle the full range of impossibility results involved in Free Choice. The paper develops the hypothesis that Free Choice is a homogeneity effect. The claim possibly A or B is defined only when A and B are homogenous with respect to their modal status, either both possible or both impossible. Paired with a notion of entailment that is sensitive to definedness conditions, this theory validates Free Choice while retaining a wide variety of classical principles except for the transitivity of entailment. The homogeneity hypothesis is implemented in two different ways, homogeneous alternative semantics and homogeneous dynamic semantics, with interestingly different consequences. (shrink)

A permutation group G on a set A is ${\kappa }$ -homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. G is ${\kappa }$ -transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$.Giving a partial answer to a question of (...) P. M. Neumann [6] we show that there is an ${\omega }$ -homogeneous but not ${\omega }$ -transitive permutation group on a cardinal ${\lambda }$ provided (i) ${\lambda } {\omega }$ we give a method to construct large ${\kappa }$ -homogeneous, but not ${\kappa }$ -transitive permutation groups. Using this method we show that there exist ${\kappa }^+$ -homogeneous, but not ${\kappa }^+$ -transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$. (shrink)

The faithfulness condition (FC) is a useful principle for inferring causal structure from statistical data. The usual motivation for the FC appeals to theorems showing that exceptions to it have probability zero, provided that some apparently reasonable assumptions obtain. However, some have objected that, the theorems notwithstanding, exceptions to the FC are probable in commonly occurring circumstances. I argue that exceptions to the FC are probable in the circumstances specified by this objection only given the presence of a condition that (...) I label homogeneity, and furthermore that this condition typically does not obtain in the FC’s intended domain of application. (shrink)

Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let be countable V‐structure which is homogeneous, simple and 1‐based. The first main result says that if is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if is “coordinatized” by a set with SU‐rank 1 and there is no definable (without parameters) nontrivial equivalence relation on (...) M with only finite classes, then is strongly interpretable in a random structure. (shrink)

We define an appropriate analog of the Morley rank in a totally transcendental homogeneous model with type diagram D. We show that if RM[p] = α then for some 1 ≤ n < ω the type p has n, but not n + 1, distinct D-extensions of rank α. This is surprising, because the proof of the statement in the first-order case depends heavily on compactness. We also show that types over (D,ℵ₀)-homogeneous models have multiplicity (Morley degree) 1.

Throughout his metaphysical writings, Sellars maintains that current microtheory, with its particulate paradigm, can never depict adequately---even in principle---a universe populated with sentient beings like us. Why not? Experience for us involves the presence of an occurrent perceptual core of ultimately homogeneous secondary qualities. Sellars’ “Grain Argument” demonstrates that physical objects qua clouds of discrete particles cannot instantiate such qualities and that they cannot be assigned to an intrasentient realm construed as clusters of discrete, particulate neurons. Neither, contends Sellars, can (...) they simply be eliminated from the inventory of any theory claiming to be both empirical and conceptually independent of common sense. And since common sense fails to provide an adequate picture of reality, our only course is to abandon the particulate paradigm of current microtheory in favor of a process paradigm. This paper traces and develops, in dialog form, these arguments. (shrink)

In his recent book Scientific Explanation and the Causal Structure of the World Wesley Salmon provides a detailed explanation of objective homogeneity, a concept which is central to his S-R model of explanation. 1 propose a modification of Salmon's definition which both simplifies and (in minor ways) corrects it, while at the same time generalizes it by including an important temporal factor that is missing from the original. I argue that if the world is irreducibly stochastic, then objective probabilities (...) (determined by objective homogeneous reference classes) must be temporally relativized. We can speak coherently of the objective probability of a particular event relative to a given point in time, but not of the objective probability of the event simpliciter. I briefly explore the consequences, of the temporal relativity of objective homogeneity for Salmon's attempt to secure an objective (nonepistemic, nonpraagmatic) S-R basis for causal explanation. (shrink)

We exploit the properties of homogeneous functions to characterize the symmetric pure-strategy Nash equilibria of n-player symmetric games in which each player’s revenue function is not homogeneous but it can be decomposed into the sum of homogeneous functions with different degrees of homogeneity. Our results aim to provide a pathway for an easy computation of symmetric equilibria for this type of games. We discuss our results in a Cournot game, a contest game, and a public good game.

Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.Following the familiar simplification of Russell's theory, let us mean by alogical typein the strict sense any one of the following: (i) (...) the totality consisting exclusively of individuals, (ii) the totality consisting exclusively of classes whose members are exclusively individuals, (iii) the totality consisting exclusively of classes whose members are exclusively classes whose members in turn are exclusively individuals, and so on. Any entity from (ii) is said to beone type higherthan any entity from (i), any entity from (iii), one type higher than any entity from (ii), and so on. In systems based upon this simplified theory of types, the only significant atomic formulae involving “ε” are those asserting the membership of an entity in an entity one type higher. Thus any expression of the form “(x∈y)” is meaningless except where “y” denotes an entity of just one type higher than the type of the entity denoted by “x” It is by means of general type restrictions of this kind that the Russell and other paradoxes are avoided. (shrink)

We present some results on countable homogeneous 3-hypergraphs. In particular, we show that there is no unexpected homogeneous 3-hypergraph determined by a single constraint.

Our world is increasingly torn between the cultural polarities of a homogenized neo-liberal globalization and a variety of exclusionary fundamentalisms. The text being discussed opens a third space of what it calls ‘convergent pluralism’ between these two, that promises to evolve a harmonious planetary future. The present article discusses the analyses and solutions offered by the book as well as the constructive attitudes and engagements needed to make this third space a reality.

A key facet of Valerie Plumwood’s feminist critique of logic is her analysis of classical negation. On Plumwood’s reading, the exclusionary features of classical negation generate hierarchical dualisms, i.e., dichotomies in which dominant groups’ primacy is reinforced while underprivileged groups are oppressed. For example, Plumwood identifies the system collapse following from ex contradictione quodlibet—that a theory including both φ and ∼φ trivializes—as a primary source of many of these features. Although Plumwood considers the principle of excluded middle to be compatible (...) with her goals, that she identifies relevant logics as systems lacking a hierarchical negation—whose first-degree fragments are both paraconsistent and paracomplete—suggests that excluded middle plays some role in hierarchical dualisms as well. In these notes, I examine the role of excluded middle in generating oppressive homogenization and try to clarify the relationship between Plumwood’s critique and this principle from several contemporary perspectives. Finally, I examine the matter of whether Plumwood’s critique requires relevance or whether a non-relevant logic could satisfy her criteria and serve as a liberatory logic of difference. (shrink)

We give an example of a simple ω-categorical theory such that for any finite set of parameters the corresponding constant expansion does not satisfy the PAPA. We describe a wide class of homogeneous structures with generic automorphisms and show that some natural reducts of our example belong to this class.

We present three examples of countable homogeneous structures whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric (...) groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free. (shrink)

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations St $${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{$2$}}$$ S x 2 =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the (...) terms of these expansions and this suggests that S(x, t) should be written as a product of the elementary solution x2/(2t) and a function f=f(φ) where φ=φ(x, |t|) owing to the symmetry property which is that S(x, −t)=−S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of φ(x, |t|)=φ0 |t/t0|κ (x/x 0)λ≥0 with κ, λ∈ ℝ results in a soluble ordinary differential equation, of first order in u=ln φ and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called “trivial.” The phase plane (fu, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the “trivial” fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid. (shrink)

Ebonics—Italian Style Usually, one has to read a book in order to evaluate it. Harney's book on “Italian” immigrants has the dubious distinction of capturing much of what is wrong with it on its cover. The very title prefigures its basic problem. Eh, Paesan! presumably refers to a particular greeting these immigrants employ in addressing each other. But there is no standard term “paesan.” The proper Italian word is “paesano,” which in its many dialectal renditions becomes “paisah,” “paisan” or variations (...) thereof. The same goes for the antecedent “Eh!” which in most Southern Italian dialects is actually “uí,” “ué,” “eeeeh,”…. (shrink)

Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model A has a d-basis if the types realized in A are all computable and the Turing degree d can list $\Delta _{0}^{0}$ -indices for all types realized in A. We say A has a d-decidable copy if there exists a model B ≅ A such that the elementary (...) diagram of B is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous A with a 0-basis but no decidable copy. We prove that any homogeneous A with a 0'-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and A is a homogeneous model with a 0-basis, we show A has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous A with a 0-basis has a d-decidable copy. We show that the nonlow₂ $\Delta _{2}^{0}$ degrees are 0-homogeneous bounding. (shrink)

We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs and determine for which first-order languages the isomorphism problem for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the complexity of the isometry relation (...) for certain classes of homogeneous and ultrahomogeneous metric spaces. (shrink)

If one examines the sky at sunset on a clear night, one seems to see a continuum of colors from reds, oranges and yellows to a deep blue-black. Between any two colored points in the sky there seem to be other colored points. Furthermore, the changes in color across the sky appear to be continuous. Although the colors at the zenith and the horizon are obviously distinct, nowhere in the sky can one see any color borders, and every sufficiently small (...) region of the sky is made up of regions that all seem to be of the same color. (shrink)