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The Mathematician as a Formalist
 in Truth in Mathematics (H.G. Dales and
, 1998
"... Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next mil ..."
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Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)
Foundations of Physics manuscript No.
, 1012
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Generalized Power Series and NonStandard Analysis Between Model Theory and NSA
"... consists of the series of the form: ant rn, n=0 where an ∈ R and r0 < r1 < r2 <... → ∞. Notice that R 〈 t R 〉 is a real closed field. R 〈 t R 〉 is a nonArchimedean field. In particular t is a positive infinitesimal, i.e. 0 < t < 1/n for all n ∈ N. The valuation is defined by v: R 〈 t ..."
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consists of the series of the form: ant rn, n=0 where an ∈ R and r0 < r1 < r2 <... → ∞. Notice that R 〈 t R 〉 is a real closed field. R 〈 t R 〉 is a nonArchimedean field. In particular t is a positive infinitesimal, i.e. 0 < t < 1/n for all n ∈ N. The valuation is defined by v: R 〈 t R 〉 → R ∪ {∞}, where v ( ∑ ∞ n=0 ant rn) = r0 if a0 = 0, and v(0) = ∞. R 〈 t R 〉 is sequentially complete (but it is neither spherically complete, nor it is Cantor complete). R ⊂ R(t) ⊂ R(t Z) ⊂ R 〈 t R 〉.
URL: www.emis.de/journals/AFA/ KAPLANSKY’S AND MICHAEL’S PROBLEMS: A SURVEY
"... Abstract. I. Kaplansky showed in 1947 that every submultiplicative norm ‖. ‖ on the algebra C(K) of complex–valued functions on an infinite compact space K satisfies ‖f ‖ ≥ ‖f‖K for every f ∈ C(K), where ‖f‖K = maxt∈Kf(t) denotes the standard norm on C(K). He asked whether all submultiplicative n ..."
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Abstract. I. Kaplansky showed in 1947 that every submultiplicative norm ‖. ‖ on the algebra C(K) of complex–valued functions on an infinite compact space K satisfies ‖f ‖ ≥ ‖f‖K for every f ∈ C(K), where ‖f‖K = maxt∈Kf(t) denotes the standard norm on C(K). He asked whether all submultiplicative norms ‖. ‖ were in fact equivalent to the standard norm (which is obviously true for finite compact spaces), or equivalently, whether all homomorphisms from C(K) into a Banach algebra were continuous. This problem turned out to be undecidable in ZFC, and we will discuss here some recent progress due to Pham and open questions concerning the structure of the set of nonmaximal prime ideals of C(K) which are closed with respect to a discontinuous submultiplicative norm on C(K) when the continuum hypothesis is assumed. We will also discuss the existence of discontinuous characters on Fréchet algebras (Michael’s problem), a long standing problem which remains unsolved. The Mittag–Leffler theorem on inverse limits of complete metric spaces plays an essential role in the literature concerning both problems. 1. Introduction and
Maximal chains of closed prime ideals for discontinuous algebra norms on
, 2013
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Foundations of Regular Variation
"... The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have two kin ..."
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The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have two kinds of common generalization, both of a combinatorial nature; one is exemplified by ‘containment up to translation of subsequences’, the other, drawn from descriptive set theory, requires nonemptiness of a Souslin ∆ 1 2set. All of our generalizations are equivalent to the uniform convergence property.
On Ordered Fields with Infinitely Many Integer Parts
"... We investigate integer parts of ordered fields. We prove the existence of normal integer parts for a class of ordered fields. Along with the normal one we construct infinitely many elementary nonequivalent integer parts for each field from this class. ..."
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We investigate integer parts of ordered fields. We prove the existence of normal integer parts for a class of ordered fields. Along with the normal one we construct infinitely many elementary nonequivalent integer parts for each field from this class.
THE KERNELS AND CONTINUITY IDEALS OF HOMOMORPHISMS FROM C0(Ω) HUNG LE PHAM
"... Abstract. We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous funct ..."
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Abstract. We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous functions. 1.
Lecture Notes: NonStandard Approach to J.F. Colombeau’s Theory of Generalized Functions
"... In these lecture notes we present an introduction to nonstandard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau ’ theory of new generalized functions and its applications. The main purpose of our nonstandard approach to ..."
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In these lecture notes we present an introduction to nonstandard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau ’ theory of new generalized functions and its applications. The main purpose of our nonstandard approach to Colombeau ’ theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our nonstandard approach the sets of scalars of the functional spaces always form algebraically closed nonarchimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau’s theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau’s theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau’s theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau’s community, the readers who are already familiar with nonstandard methods might also find a short and comfortable way to learn about Colombeau’s theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differential equations, differential geometry, relativity theory and other areas of mathematics and physics.