Veblen Hierarchy

The Veblen hierarchy is an extension of the construction of epsilon numbers (fixpoints of the exponential map: ωε = ε). It is a collection φα of the Veblen Functions where φ0(β) = ωβ and φ1(β) = εβ. The sequence of fixpoints of φ1 function form φ2, etc. For a limit non empty ordinal λ the function φλ is the sequence of common fixpoints of all functions φα where α < λ. The Mizar formalization of the concept cannot be done directly as the Veblen functions are classes (not (small) sets). It is done with use of universal sets (Tarski classes). Namely, we define the Veblen functions in a given universal set and φα(β) as a value of Veblen function from the smallest universal set including α and β.Białystok Technical University, PolandGrzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. Köonig's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Grzegorz Bancerek. Tarski's classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123-129, 1990.Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265-267, 1990.Grzegorz Bancerek. Epsilon numbers and Cantor normal form. Formalized Mathematics, 17(4):249-256, 2009, doi: 10.2478/v10037-009-0032-8.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595-600, 1990.Karol Pąk. The Nagata-Smirnov theorem. Part I. Formalized Mathematics, 12(3):341-346, 2004.Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990

An Upper Bound on the Complexity of Recognizable Tree Languages

The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^\omega$ into the class ${\bf\Delta}^1\_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bf\Delta}^1\_2$

Hyperations, Veblen progressions and transfinite iterations of ordinal functions

In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation of a normal function f is a sequence of normal functions so that f^0= id, f^1 = f and for all ordinals \alpha, \beta we have that f^(\alpha + \beta) = f^\alpha f^\beta. These conditions do not determine f^\alpha uniquely; in addition, we require that the functions be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given \alpha, \beta, f^(\alpha + \beta)= f^\beta f^\alpha. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study cohyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory

The Strength of the Veblenian Critique of Neoclassical Economics

More than one hundred years ago, Thorstein Veblen wrote a powerful critique of neoclassical economics that castigated the discipline for turning the individual into a “lightning calculator of pleasures and pains, who oscillates like a homogeneous globule”, or equivalently, for the individual’s static maximization of utility based on exogenous preferences. His critique is relevant even today, since there are economists who still continue to criticize the assumptions of homo economicus and exogenous preferences, and insist on introducing more realism to economic theory. Furthermore, recent developments in game theory and experimental economics, which stand at the cutting-edge of economics today, are far more accommodating to the ideas of institutions that were central to Veblen’s theory than neoclassical economics. The goal of this paper is to examine the strengths of the Veblenian critique of neoclassical economics. In particular, it investigates whether or not Veblen’s rejection of the axiomatic approaches to economics is merely an attack on neoclassical economics which fails to provide an alternative positive theory. Starting with their conception of the individual, going through their theoretical frameworks, and ending with an investigation of how they approach a concrete issue, this paper offers a comparative exposition of the Veblenian and neoclassical approaches to economic theory. [excerpt

A Cognitive and Social Psychological Perspective on the Demand for Fashion

The fashion industry is an important global industry. In 2012, in Britain it generated more than £48 billion annually and employs more than 600,000 people. Whether or not we consider ourselves fashionable or interested in fashion, we all clothe ourselves in items we have bought or have chosen to wear. Individuals in developed countries tend to buy more fashion items than they need and many have bought items that remain unworn. The fashion industry depends on demand for new ideas and products which allow individuals to perceive themselves as socially or economically superior or simply different. From a psychological perspective, this is in conflict with the well understood desire to conform. The psychological underpinnings of the demand for fashion are complex and have been neglected in research. This paper considers the cognitive and social psychological roles of decision making in the demand for fashion

The role of routines, rules and habits in collective learning: some epistemological and ontological considerations

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Veblen in the (Inner) City: On the Normality of Looting

Drawing on Veblen's concept of 'pecuniary prowess' I will argue that the August riots can be understood not so much in terms of protest but as an appropriation of the underlying acquisitive logic of capitalism. The violent realisation of that logic across class divides has become more likely due to an erosion in plausibility of discourses of meritocratic legitimacy. Recent denigrating discourses around \"chavs\" as dangerous and undeserving poor can be understood as attempts to reinstate meritocratic legitimacy rhetorically, but in an increasingly unequal society this becomes an ever more difficult enterprise. On the other hand, the assertion of the order of property through an effective police response may have eased the pressure by providing evidence that anxieties about a full scale insurgence are unfounded.August 2011 Riots; Thorstein Veblen; Inequality; Capitalism; Violence

Post-Keynesian Theory of Business Enterprise and the Veblenian´s Approach: Are there commonalities?

The main objective of this paper is to explore the possible common grounds, divergences and complementarities between the Veblenian’s approach on the Theory of Business Enterprise followed by Institutional economists, and the modern Post Keynesian Micro theory on Business enterprise. Due to the dispersion and lack of systematization of Institutional Economics regarding this body of theory, compared with the Post-Keynesian theory of the firm, the main efforts of this paper will be dedicated to a short survey of the Institutional approach. In the second section of the paper I review the basic ideas presented in Veblen’s main contributions on this area regarding business enterprises (industrial process, main principles, role of credit, ownership structure, the legal framework, the price behavior and the cultural incidences). Then, I make a comparison with the main theoretical results that modern postkeynesian vision has developed regarding structure of production, costing, pricing, investment, and competition and market governance . I claim that even though there are commonalities and some minor divergences between the two approaches, complementarities among them are more relevant, although the main areas of research have been somewhat different. I end with some conclusions that underline possible areas of cooperation between these two schools of economic thought within the heterodox paradigm.Theory of the Firm, Postkeynesian Economics, Institutional Economics.