17 found
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  1.  8
    Computability Theory, Nonstandard Analysis, and Their Connections.Dag Normann & Sam Sanders - 2019 - Journal of Symbolic Logic 84 (4):1422-1465.
    We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a (...)
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  2.  9
    Erna and Friedman's Reverse Mathematics.Sam Sanders - 2011 - Journal of Symbolic Logic 76 (2):637 - 664.
    Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA₀ from Reverse Mathematics (see [21] and [22]) can be 'pushed down' (...)
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  3.  21
    Transfer and a Supremum Principle for ERNA.Chris Impens & Sam Sanders - 2008 - Journal of Symbolic Logic 73 (2):689 - 710.
    Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Patrick Suppes and Richard Sommer, who also proved its consistency inside PRA. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes, of which Michal Rössler and Emil Jeřábek have recently proposed a weakened version. We add a Π₁-transfer principle to ERNA and prove the consistency of the extended theory inside PRA. In this extension of ERNA a σ₁-supremum (...)
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  4.  33
    More Infinity for a Better Finitism.Sam Sanders - 2010 - Annals of Pure and Applied Logic 161 (12):1525-1540.
    Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed in around 1995 by Patrick Suppes and Richard Sommer. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes. Here, we discuss the inherent problems and limitations of the classical nonstandard framework and propose a much-needed refinement of ERNA, called , in the spirit of Karel Hrbacek’s stratified set theory. We study the metamathematics of and its (...)
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  5.  9
    Pincherle's Theorem in Reverse Mathematics and Computability Theory.Dag Normann & Sam Sanders - 2020 - Annals of Pure and Applied Logic 171 (5):102788.
    We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to (...)
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  6.  13
    Reverse Formalism 16.Sam Sanders - 2020 - Synthese 197 (2):497-544.
    In his remarkable paper Formalism 64, Robinson defends his eponymous position concerning the foundations of mathematics, as follows:Any mention of infinite totalities is literally meaningless.We should act as if infinite totalities really existed. Being the originator of Nonstandard Analysis, it stands to reason that Robinson would have often been faced with the opposing position that ‘some infinite totalities are more meaningful than others’, the textbook example being that of infinitesimals. For instance, Bishop and Connes have made such claims regarding infinitesimals, (...)
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  7.  31
    The Dirac Delta Function in Two Settings of Reverse Mathematics.Sam Sanders & Keita Yokoyama - 2012 - Archive for Mathematical Logic 51 (1-2):99-121.
    The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property ${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$ of the Dirac delta function. We show that (...)
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  8.  7
    Splittings and Disjunctions in Reverse Mathematics.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (1):51-74.
    Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with two RM-phenomena, namely, splittings and disjunctions. As to splittings, there are some examples in RM of theorems A, B, C such that A↔, that is, A can be split into two (...)
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  9.  17
    Reverse-Engineering Reverse Mathematics.Sam Sanders - 2013 - Annals of Pure and Applied Logic 164 (5):528-541.
    An important open problem in Reverse Mathematics is the reduction of the first-order strength of the base theory from IΣ1IΣ1 to IΔ0+expIΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+expIΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity . The logical principle (...)
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  10.  13
    The Strength of Compactness in Computability Theory and Nonstandard Analysis.Dag Normann & Sam Sanders - 2019 - Annals of Pure and Applied Logic 170 (11):102710.
  11.  24
    On the Mathematical and Foundational Significance of the Uncountable.Dag Normann & Sam Sanders - 2019 - Journal of Mathematical Logic 19 (1):1950001.
    We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...)
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  12.  17
    A Note on Non-Classical Nonstandard Arithmetic.Sam Sanders - 2019 - Annals of Pure and Applied Logic 170 (4):427-445.
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  13.  15
    Reverse Mathematics and Parameter-Free Transfer.Benno van den Berg & Sam Sanders - 2019 - Annals of Pure and Applied Logic 170 (3):273-296.
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  14.  11
    Refining the Taming of the Reverse Mathematics Zoo.Sam Sanders - 2018 - Notre Dame Journal of Formal Logic 59 (4):579-597.
    Reverse mathematics is a program in the foundations of mathematics. It provides an elegant classification in which the majority of theorems of ordinary mathematics fall into only five categories, based on the “big five” logical systems. Recently, a lot of effort has been directed toward finding exceptional theorems, that is, those which fall outside the big five. The so-called reverse mathematics zoo is a collection of such exceptional theorems. It was previously shown that a number of uniform versions of the (...)
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  15.  25
    On Algorithm and Robustness in a Non-Standard Sense.Sam Sanders - 2013 - In Hanne Andersen, Dennis Dieks, Wenceslao González, Thomas Uebel & Gregory Wheeler (eds.), New Challenges to Philosophy of Science. Springer Verlag. pp. 99--112.
  16.  29
    On the Connection Between Nonstandard Analysis and Constructive Analysis.Sam Sanders - forthcoming - Logique Et Analyse.
    Constructive Analysis and Nonstandard Analysis are often characterized as completely antipodal approaches to analysis. We discuss the possibility of capturing the central notion of Constructive Analysis (i.e. algorithm, finite procedure or explicit construction) by a simple concept inside Nonstandard Analysis. To this end, we introduce Omega-invariance and argue that it partially satisfies our goal. Our results provide a dual approach to Erik Palmgren's development of Nonstandard Analysis inside constructive mathematics.
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  17.  18
    Relative Arithmetic.Sam Sanders - 2010 - Mathematical Logic Quarterly 56 (6):564-572.
    In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y -standard’. Thus, objects are not standard in an absolute sense, but standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical (...)
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