J. Barkley Rosser

Extensions of some theorems of Gödel and Church

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Godel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Godel calls “rekursiv” will be called “primitive recursive.” By an “ Entscheidungsverfahren ” will be meant a general recursive function ϕ ( n ) such that, if n is the Godel number of a provable formula, ϕ ( n ) = 0 and, if n is not the Godel number of a provable formula, ϕ ( n ) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.” First, a modification is made in Godels proofs of his theorems, Satz VI (Godel, p. 187—this is the theorem which states that ω -consistency implies the existence of undecidable propositions) and Satz XI (Godel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Godels Satz VI and Kleenes Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).

Sharper Bounds for the Chebyshev Functions Theta(x) and Psi(x).

Abstract : The authors demonstrate a wider zero-free region for the Riemann zeta function than has been given before. They give improved methods for using this and a recent determination that 3,500,000 zeros lie on the critical line to develop better bounds for functions of primes.

Complex Ecologic-Economic Dynamics and Environmental Policy

Various complex dynamics in ecologic-economic systems are presented with an emphasis upon models of global warming and fishery dynamics. Chaotic and catastrophic dynamic patterns are shown to be possible, along with amplified oscillations due to these non-linear interactions in the combined interactions of human economic decisionmaking with ecological dynamics are identified and discussed. Implications for policy are examined with strong recommendations for greater emphasis in particular upon the precautionary principle to avoid catastrophic collapses beyond critical thresholds and the scale-matching principle to ensure that efforts to manage complex non-linear dynamics are directed at the appropriate levels of ecologic-economic interaction.

The Complexity Era in Economics

This article argues that the neoclassical era in economics has ended and is being replaced by a new era. What best characterizes the new era is its acceptance that the economy is complex, and thus that it might be called the complexity era. The complexity era has not arrived through a revolution. Instead, it has evolved out of the many strains of neoclassical work, along with work done by less orthodox mainstream and heterodox economists. It is only in its beginning stages. The article discusses the work that is forming the foundation of the complexity era, and how that work will likely change the way in which we understand economic phenomena and the economics profession.

Highlights of the History of the Lambda-Calculus

This paper gives an account of both the lambda-calculus and its close relative, the combinatory calculus, and explains why they are of such importance for computer software. The account includes the shortest and simplest proof of the Church-Rosser theorem, which appeared in a limited printing in August 1982. It includes a model of the combinatory calculus, also available in 1982 in a limited printing. In the last half-dozen years, some revolutionary new ideas for programming have appeared, involving the very fundamentals of the lambda-calculus and the combinatory calculus. A short introduction is given for a couple of these new ideas.

Econophysics And Economic Complexity

This paper discusses the debate between those advocating a computational and those advocating a dynamic definition of complexity, and how this relates to issues in econophysics. It then reviews the criticisms that have been raised about ways in which econophysics has been done, noting that many of these are now being dealt with. Finally, it argues that while an obvious way to resolve many of these matters is to have economists and physicists work together, the physicists should be sure to work with economists who understand the complexity critique of conventional economic theory and are thus not led astray into building models that have some of the problems of standard economics models that most econophysicists have striven to overcome.

An Informal Exposition of Proofs of Godel's Theorems and Church's Theorem

This paper is an attempt to explain as non-technically as possible the principles and devices used in the various proofs of Godels Theorems and Churchs Theorem. Roman numerals in references shall refer to the papers in the bibliography. In the statements of Godels Theorems and Churchs Theorem, we will employ the phrase “for suitable L .” The hidden assumptions which we denote by this phrase have never been put down explicitly in a form intelligible to the average reader. The necessity for thus formulating them has commonly been avoided by proving the theorems for special logics and then remarking that the proofs can be extended to other logics. Hence the conditions necessary for the proofs of Godels Theorems and Churchs Theorem are at present very indefinite as far as the average reader is concerned. To partly clarify this situation, we will now mention the more prominent of these assumptions. I. In any proof of Godels Theorems or Churchs Theorem, two logics are concerned. One serves as the “logic of ordinary discourse” in which the proof is carried out, and the other is a formal logic, L , about which the theorem is proved. The first logic may or may not be formal. However L must be formal. Among other things, this implies that the propositions of L are formulas built according to certain rules of structure.

All That I Have to Say Has Already Crossed Your Mind

We present three arguments regarding the limits to rationality, prediction and control in economics, based on Morgensterns analysis of the Holmes-Moriarty problem. The first uses a standard metamathematical theorem on computability to indicate logical limits to forecasting the future. The second provides possible nonconvergence for Bayesian forecasting in infinite-dimensional space. The third shows the impossibility of a computer perfectly forecasting an economy with agents knowing its forecasting program. Thus, economic order is partly the product of something other than calculative rationality. The joint presentation of these existing results should introduce the reader to implications of these concepts for certain shared concerns of Keynes and Hayek.

The dialogue between the economic and the ecologic theories of evolution

Abstract The economic and ecologic theories of evolution have developed by means of a mutual dialogue and interaction from the early nineteenth century to the present, despite certain fundamental differences between them. This mutual influence has involved the rise of the equilibrium concept as well as the appearance of cyclical and chaotic dynamic models. Both theories have faced a conflict between gradualistic and punctuationist (saltationalist) approaches. Both have emphasized the emergence of higher-order self-reproducing structures, a process of hypercyclic morphogenesis ultimately encouraged by the dialogue itself.

The Development of Complex Oligopoly Dynamics Theory

This chapter will review the development of the theory of complex oligopoly dynamics from the 1970s to the year 2001 in its main strands. It will also provide certain speculations regarding possible future developments. This will serve as a link between the first chapter’s discussion of the broader history of oligoply theory up to the 1940s and the more specific chapters that present current models in the rest of this book. But, to tell our story we first need to remind ourselves of certain ideas from the first chapter of this book. One is the very founding document of oligopoly theory, Cournot’s seminal work of 1838. This is both because the specific model that he presented has been much studied for its ability to generate complex dynamics and also because of its more general foreshadowing of game theory. It has often been noted that the Cournot equilibrium is but a special case of the Nash (1951) equilibrium, the more general formulation used by modern industrial organization economists in studying oligopoly theory. Indeed, it is sometimes even called the Cournot-Nash equilibrium. Although many of the models of complex oligopoly dynamics use the specific Cournot model, many use more general game theoretic formulations. We note simply as an aside here that Cournot’s work was the first to apply calculus to solving an economic optimization problem and also was the first to introduce supply and demand curves, albeit in the “Walrasian” form with price on the horizontal axis.