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Real Numbers Real analysis studies more pathological functions than complex analysis, so more secure footing was necessary. Notion of (positive) real number arose early as a ratio of lengths. Existence of transcendental (non-algebraic) real numbers: 1 / 10 + 1 / 10 2 + 1 / 10 6 + 1 / 10 24 +··· (Liouville, 1851) e (Hermite, 1873) π (Lindemann, 1882) Dedekind’s geometric construction published in 1878. Dedekind’s method had roots in work done 1500 years earlier by Eudoxus: D 1 : S 1 = D 2 : S 2 iF for every m,n N mD 1 < = > mD 2 nS 1 < = > nS 2 . Cantor’s analytic construction (about 1872) What’s wrong with axiomatic approach? What’s wrong with in±nite decimals?
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Unformatted text preview: Infnite Ordinals and Cardinals • ²or ±nite ordinals and cardinals, the distinction is linguistic and conceptual, not mathemat-ical. • Trans±nite ordinal numbers were introduced by Cantor in 1883. Order types of well-ordered sets. Noncommutative arithmetic. (More to come on the reason for their discovery.) • Trans±nite cardinal numbers were introduced by Cantor in 1895. Some algebraic rules. Hyperreal Numbers • More or less invented by Abraham Robinson in 1960s to allow working with diFerentials in a rigorous matter. “Nonstandard analysis.” Not widely used. (Maybe more to come.)...
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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