Infinitesimal Methods in Mathematical Economics
Infinitesimal Methods in Mathematical Economics Robert M. Anderson1 Department of Economics and Department of Mathematics University of California at Berkeley Berkeley, CA 94720, U.S.A. and Department of Economics Johns Hopkins University Baltimore, MD 21218, U.S.A. January 20, 2008 1The author is grateful to Marc Bettz¨uge, Don Brown, Hung- Wen Chang, G´erard Debreu, Eddie Dekel-Tabak, Greg Engl, Dmitri Ivanov, Jerry Keisler, Peter Loeb, Paul MacMillan, Mike Magill, Andreu Mas-Colell, Max Stinchcombe, Cathy Wein- berger and Bill Zame for their helpful comments. Financial sup- port from Deutsche Forschungsgemeinschaft, Gottfried-Wilhelm- Leibniz-F¨orderpreis is gratefully acknowledged. Contents 0Preface v 1 Nonstandard Analysis Lite 1 1.1 When is Nonstandard Analysis Useful? . 1 1.1.1 Large Economies . 2 1.1.2 Continuum of Random Variables . 4 1.1.3 Searching For Elementary Proofs . 4 1.2IdealElements................. 5 1.3Ultraproducts.................. 6 1.4InternalandExternalSets........... 9 1.5NotationalConventions............. 11 1.6StandardModels................ 12 1.7 Superstructure Embeddings . 14 1.8AFormalLanguage............... 16 1.9TransferPrinciple................ 16 1.10Saturation.................... 18 1.11InternalDefinitionPrinciple.......... 19 1.12 Nonstandard Extensions, or Enough Already with the Ultraproducts . 20 1.13HyperfiniteSets................. 21 1.14 Nonstandard Theorems Have Standard Proofs 22 2 Nonstandard Analysis Regular 23 2.1 Warning: Do Not Read this Chapter . 23 i ii CONTENTS 2.2AFormalLanguage............... 23 2.3 Extensions of L ................. 25 2.4 Assigning Truth Value to Formulas . 28 2.5 Interpreting Formulas in Superstructure Em- beddings..................... 32 2.6TransferPrinciple................ 33 2.7InternalDefinitionPrinciple.......... 34 2.8NonstandardExtensions............ 35 2.9TheExternalLanguage............. 36 2.10TheConservationPrinciple.......... 37 3RealAnalysis 39 3.1Monads..................... 39 3.2OpenandClosedSets............. 44 3.3Compactness.................
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