1,721,029 research outputs found
Note in margine a 'Kant und die moderne Mathematik' di Ernst Cassirer
No full textI consider Ernst Cassirer's views in his paper 'Kant und die moderne Mathematik' (1907)
in the light of some developments in logic and the foundations of mathematics in the Twentieth Century
On the consistency of ZF set theory and its large cardinal extensions
No full textI consider the question of the consistency of ZF set theory and of its large cardinal extensions, from both a historical and a theoretical point of view, touching on some epistemological aspects of the problem. First I recall the reasons why in the case of set theory neither model- theoretic nor proof-theoretic methods seem suitable for the question of consistency. Then I show how set-theorists have dealt with the problem by means of large cardinals and inner models, with a remarkable confidence in a sort of direct intuition of consistency. I argue that the relationship between intuition and formalization is the crucial point regarding the metamathematical treatment of the problem. Finally, I show how finitary versions of Goedel sentence constructions due to W. H. Woodin could give evidence that there is no substantial dissimilarity, from an epistemological point of view, between the problems of consistency for arithmetic and for large cardinal hypotheses. The ideas of Skolem, Zermelo, Goedel, Kreisel and Cohen on the topic are touched upon
Decoding Gentzen's notation
No full textIn this note we consider Gentzen's first ordinal notation, used in his first published proof of the consistency of Peano Arithmetic (1936). It is a decimal notation, quite different from our current notations. We give a rule to translate this notation into our usual set-theoretic notation and we show some of its peculiarities. Then we indicate how to decode Gentzen's assignment of ordinal notations to derivations and give some examples. Finally, we go through his proof of their decrease after the application of his reduction procedure, giving further examples
Dalla Rivoluzione a Stalin: la lotta per la logica matematica in Unione Sovietica (Pristem/Storia 43)
No full textIl volume contiene un breve saggio sullo sviluppo della logica matematica in Unione Sovietica dal 1917 al 1947 e la prima traduzione dal russo in una lingua occidentale del lavoro di S. A. Yanovskaya 'Fondamenti della matematica e logica matematica' (1948)
On the circularity of set-theoretic semantics for set theory
No full textThe set-theoretic nature of the usual semantics of set theory raises a problem of circularity. A recourse to an intuitive semantics (possibly in terms of the iterative concept of set) is often deemed necessary, and a certain kind of realist philosophy of mathematics seems its best justification, taking for granted a well-determined reality of which set-theoretic statements are true. I argue that, on the contrary, this form of realism leaves one in even deeper trouble. I try to understand the circularity of the set-theoretic semantics of set theory and the related crucial problem of quantification over the universe of sets in the light of a different, 'Neo-Kantian' perspective
Formalization, syntax, and the standard model of arithmetic
No full textI make an attempt at the description of the delicate role of the standard model of arithmetic for the syntax of formal systems. I try to assess whether the possible instability in the notion of finiteness deriving from the nonstandard interpretability of arithmetic affects the very notions of syntactic metatheory and of formal system. I maintain that the crucial point of the whole question lies in the evaluation of the phenomenon of formalization. The ideas of Skolem, Zermelo, Beth and Carnap (among others) on the problem are discussed
Von Neumann's consistency proof
Get PDFWe consider the consistency proof for a weak fragment of arithmetic published by von Neumann in 1927. This proof is rather neglected in the literature on the history of consistency proofs in the Hilbert school. We explain von Neumann’s proof and argue that it fills a gap between Hilbert’s consistency proofs for the so-called elementary calculus of free variables with a successor and a predecessor function and Ackermann’s consistency proof for second-order primitive recursive arithmetic. In particular, von Neumann’s proof is the first rigorous proof of the consistency of an axiomatization of the first-order theory of a successor function
Oggetti matematici: verità e conoscenza. Sul dilemma di Benacerraf
No full textIn this note I discuss Benacerraf's dilemma in the philosophy of mathematics (on knowability and truth of mathematical sentences), some possible reactions, and an attempt at a solution
Woodin on the Continuum Problem: an overview and some objections
No full textI consider W. Hugh Woodin's approach to the Continuum Problem. I summarize Woodin's
main results and then discuss some objections which have been raised to his
approach
Logica aristotelica e logica moderna
No full textIn this note I give a brief account of the relationship between Aristotle's logic and modern (Fregean) logic
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