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Ultraproducts in Analysis
 IN ANALYSIS AND LOGIC, VOLUME 262 OF LONDON MATHEMATICAL SOCIETY LECTURE NOTES
, 2002
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HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS
, 708
"... Abstract. We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilb ..."
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Abstract. We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert’s fifth problem for global groups by Hirschfeld. 1.
1 Ends of groups: a nonstandard perspective
"... Abstract: We give a nonstandard treatment of the notion of ends of proper geodesic metric spaces. We then apply this nonstandard treatment to Cayley graphs of finitely generated groups and give nonstandard proofs of many of the fundamental results concerning ends of groups. We end with an analogous ..."
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Abstract: We give a nonstandard treatment of the notion of ends of proper geodesic metric spaces. We then apply this nonstandard treatment to Cayley graphs of finitely generated groups and give nonstandard proofs of many of the fundamental results concerning ends of groups. We end with an analogous nonstandard treatment of the ends of relatively Cayley graphs, that is Cayley graphs of cosets of finitely generated groups.
NONSTANDARD ANALYSIS IN POINTSET TOPOLOGY
"... Abstract We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to pointset topology. Some of the topological topics such as th ..."
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Abstract We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to pointset topology. Some of the topological topics such as the Hewitt realcompactification and the nonstandard characterization of the sober spaces seem to be new in the literature on nonstandard analysis. Others have already close counterparts but they are presented here with essential simplifications.
Rolle leaves and ominimal structures
, 2006
"... Let R be a real closed field, and let R be a (modeltheoretic) expansion of R with the intermediate value property (IVP). We develop a version of Khovanskii theory relative to R over an ominimal expansion of R. We also introduce a notion of the relative Pfaffian closureP(R̃,R) of an ominimal str ..."
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Let R be a real closed field, and let R be a (modeltheoretic) expansion of R with the intermediate value property (IVP). We develop a version of Khovanskii theory relative to R over an ominimal expansion of R. We also introduce a notion of the relative Pfaffian closureP(R̃,R) of an ominimal structure R ̃ with respect to an expansionR that has the IVP. We prove thatP(R̃,R) is ominimal when R is a model of the real projective hierarchy. Using this result, we obtain a strong uniformity result on definable sets in P(R̃), the Pfaffian closure of an ominimal expansion R ̃ of the real field.
Nonstandard hulls of locally uniform groups, arXiv:1203.6593v1 [math.LO
, 2012
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Philosophy and Model Theory Plurals, Predicates, and Paradox Research Seminar Autumn 2011
, 2011
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Nonstandard Analysis and Applications UC Davis Mathematics StudentRun Seminar Presentation notes
, 2006
"... Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alterna ..."
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Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alternative rigorous study of calculus beyond the limits of real analysis. In 1961, Robinson constructed the “hyperreal line ” as a direct consequence of the compactness theorem of first order logic. We will examine some typical proofs of known statements in advanced calculus and extend the nonstandard framework to other mathematical fields. 1 Introduction to FirstOrder Logic 1.1 FirstOrder Languages Define a firstorder language to be a set of symbols, as a base containing a symbol for the logical NAND, quantifiers ∃ and ∀, equality (=), grouping parenthesies/brackets, and variables (as many as are needed). Though NAND is all