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Growing commas –a study of sequentiality and concatenation. Logic Group Preprint Series 257
 Department of Philosophy, Utrecht University
, 2007
"... In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding. ..."
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In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding.
An Epistemological Approach to the Design of Training Courses on Logic
"... Introduction Mathematical logic helps to form the rational basis of common sense and, at the same time, it clashes with it. This conflict can be explained by observing that results sistematically obtained by formal logic alter deeply the rationality categories socially accepted; in this respect, it ..."
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Introduction Mathematical logic helps to form the rational basis of common sense and, at the same time, it clashes with it. This conflict can be explained by observing that results sistematically obtained by formal logic alter deeply the rationality categories socially accepted; in this respect, it is worthwhile to note that these results often express innovations of natural science, already recognized by technology, that people uses without awareness. Thus, logic is a powerful educational tool to uptodate common sense rationality, that is to transfer new paradigms of thinking. Notwithstanding this fact, mathematical logic was not given a central role in the Italian high school curriculum till few time ago. At present, the situation is changing, as the diffusion of computer science and its technology is leading to the renewal of high school curricula. This renewal recognizes the increasing importance of logic in various scientific and humanistic fields and tak
WHY THE THEORY R IS SPECIAL
, 2009
"... Abstract. Is it possible to give coordinatefree characterizations of salient theories? Such characterizations would always involve some notion of sameness of theories: we want to describe a theory modulo a notion of sameness, without having to give an axiomatization in a specific language. Such a c ..."
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Abstract. Is it possible to give coordinatefree characterizations of salient theories? Such characterizations would always involve some notion of sameness of theories: we want to describe a theory modulo a notion of sameness, without having to give an axiomatization in a specific language. Such a characterization could, e.g., be a first order formula in the language of partial preorderings that describes uniquely a degree in a particular structure of degrees of interpretability. Our theory would be contained in this degree. There are very few examples currently known along these lines, except some rather trivial ones. In this paper we provide a nontrivial characterization of TarskiMostowskiRobinson’s theory R. The characterization is in terms of the double degree structure of RE degrees of local and global interpretability. Consider the RE degrees of global interpretability that are in the minimal RE degree of local interpretability. These are the global degrees of the RE locally finitely satisfiable theories. We show that these degrees have a maximum and that R is in that maximum. In more mundane terms: an RE theory is locally finite iff it is globally interpretable in R. Dedicated to Harvey Friedman on the occasion of his 60th birthday. 1.
CARDINAL ARITHMETIC IN THE STYLE OF
"... Abstract. In this paper we show how to interpret Robinson’s Arithmetic Q and the theory R of Tarski, Mostowski and Robinson as theories of cardinals in very weak theories of relations over a domain. Bei der Verfolgung eines Hasen wollte ich mit meinem Pferd über einen Morast setzen. Mitten im Sprung ..."
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Abstract. In this paper we show how to interpret Robinson’s Arithmetic Q and the theory R of Tarski, Mostowski and Robinson as theories of cardinals in very weak theories of relations over a domain. Bei der Verfolgung eines Hasen wollte ich mit meinem Pferd über einen Morast setzen. Mitten im Sprung musste ich erkennen, dass der Morast viel breiter war, als ich anfänglich eingeschätzt hatte. Schwebend in der Luft wendete ich daher wieder um, wo ich hergekommen war, um einen größeren Anlauf zu nehmen. Gleichwohl sprang ich zum zweiten Mal noch zu kurz und fiel nicht weit vom anderen Ufer bis an den Hals in den Morast. Hier hätte ich unfehlbar umkommen müssen, wenn nicht die Stärke meines Armes mich an meinem eigenen Haarzopf, samt dem Pferd, welches ich fest zwischen meine Knie schloss, wieder herausgezogen hätte. Baron von Münchhausen.
CONCATENATION AS A BASIS FOR Q AND THE INTUITIONISTIC VARIANT OF NELSON’S CLASSIC RESULT
, 2008
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Book Review Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures
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Isle of Consciousness. According to Penrose
"... $UK 16.99 There was a time when cultured Englishmen would embark on a Grand Tour of Europe, visiting important cities and inspecting monuments and vistas. The book of Penrose is a Grand ..."
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$UK 16.99 There was a time when cultured Englishmen would embark on a Grand Tour of Europe, visiting important cities and inspecting monuments and vistas. The book of Penrose is a Grand
HUME’S PRINCIPLE, BEGINNINGS
, 2010
"... Abstract. In this note we derive Robinson’s Arithmetic from Hume’s Principle in the context of very weak theories of classes and relations. ..."
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Abstract. In this note we derive Robinson’s Arithmetic from Hume’s Principle in the context of very weak theories of classes and relations.
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?
"... Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. ..."
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Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.
STRICT PREDICATIVITY 1
"... The most basic notion of impredicativity applies to specifications or definitions of sets or classes. If a set b is specified as {x: A(x)} for some predicate A, then the specification is impredicative if A contains quantifiers such that the set b falls in the range of these quantifiers. Already in t ..."
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The most basic notion of impredicativity applies to specifications or definitions of sets or classes. If a set b is specified as {x: A(x)} for some predicate A, then the specification is impredicative if A contains quantifiers such that the set b falls in the range of these quantifiers. Already in the early history of the notion, it was extended to other cases, such as propositions in Russell’s discussions of the liar paradox. Mathematics will be predicative if it avoids impredicative definitions. The logicist reduction of the concept of natural number met a difficulty on this point, since the definition of ‘natural number ’ already given in the work of Frege and Dedekind is impredicative. More recently, it has been argued by Michael Dummett, the author, and Edward Nelson that more informal explanations of the concept of natural number are impredicative as well. That has the consequence that impredicativity is more pervasive in mathematics, and appears at lower levels, than the earlier debates about the issue generally presupposed. The appearance to the contrary resulted historically from the fact that many opponents of impredicative methods, in particular Poincaré and Weyl, were prepared to assume the natural numbers in their work. In this they were followed by later analysts of predicativity, in particular Kreisel, Schütte, and Feferman. The result was that the working conception of predicativity was of predicativity given the natural numbers. Thus Feferman characterized “the predicative conception ” as holding that “only the natural numbers can be