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  1. Internality, transfer, and infinitesimal modeling of infinite processes†.Emanuele Bottazzi & Mikhail G. Katz - forthcoming - Philosophia Mathematica.
    ABSTRACTA probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields may have advantages over hyperreals in probabilistic modeling. (...)
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  • Full algebra of generalized functions and non-standard asymptotic analysis.Todor D. Todorov & Hans Vernaeve - 2008 - Logic and Analysis 1 (3-4):205-234.
    We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between (...)
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  • Nonstandard methods in combinatorics and theoretical computer science.M. M. Richter & M. E. Szabo - 1988 - Studia Logica 47 (3):181 - 191.
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  • Debates about infinity in mathematics around 1890: The Cantor-Veronese controversy, its origins and its outcome.Detlef Laugwitz - 2002 - NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin 10 (1-3):102-126.
    This article was found among the papers left by Prof. Laugwitz (May 5, 1932–April 17, 2000). The following abstract is extracted from a lecture he gave at the Fourth Austrain Symposion on the History of Mathematics (Neuhofen/ybbs, November 10, 1995).About 100 years ago, the Cantor-Veronese controversy found wide interest and lasted for more than 20 years. It is concerned with “actual infinity” in mathematics. Cantor, supported by Peano and others, believed to have shown the non-existence of infinitely small quantities, and (...)
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  • Given a divisible ordered abelian group Λ, we call (X, d) a Λ-metric space if d: X× X−→ Λ satisfies the usual axioms of a metric, ie, for all x, y∈ X, d (x, y)− d (y, x)≥ 0 if and only if x= y, and the triangle inequality holds. We can now give the definition of asymptotic cone according to van den Dries and Wilkie.Linus Kramer & Katrin Tent - 2004 - Bulletin of Symbolic Logic 10 (2):175-185.
    §1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers (...)
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  • Asymptotic cones and ultrapowers of lie groups.Linus Kramer & Katrin Tent - 2004 - Bulletin of Symbolic Logic 10 (2):175-185.
    §1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers (...)
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  • Bounded polynomials and holomorphic mappings between convex subrings of.Adel Khalfallah & Siegmund Kosarew - 2018 - Journal of Symbolic Logic 83 (1):372-384.
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  • The Mathematical Intelligencer Flunks the Olympics.Alexander E. Gutman, Mikhail G. Katz, Taras S. Kudryk & Semen S. Kutateladze - 2017 - Foundations of Science 22 (3):539-555.
    The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and (...)
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