Lecture Notes on Nonstandard Analysis Ucla Summer School in Logic
نویسندگان
چکیده
1. The hyperreals 3 1.1. Basic facts about the ordered real field 3 1.2. The nonstandard extension 4 1.3. Arithmetic in the hyperreals 5 1.4. The structure of N∗ 7 1.5. More practice with transfer 8 1.6. Problems 9 2. Logical formalisms for nonstandard extensions 10 2.1. Approach 1: The compactness theorem 11 2.2. Approach 2: The ultrapower construction 12 2.3. Problems 16 3. Sequences and series 17 3.1. First results about sequences 17 3.2. Cluster points 19 3.3. Series 21 3.4. Problems 22 4. Continuity 23 4.1. First results about continuity 23 4.2. Uniform continuity 25 4.3. Sequences of functions 27 4.4. Problems 30 5. Differentiation 33 5.1. The derivative 33 5.2. Continuous differentiability 35 5.3. Problems 36 6. Riemann Integration 38 6.1. Hyperfinite Riemann sums and integrability 38 6.2. The Peano Existence Theorem 41 6.3. Problems 43 7. Weekend Problem Set #1 44 8. Many-sorted and Higher-Type Structures 47 8.1. Many-sorted structures 47
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