history-page8 - Infnite Ordinals and Cardinals or nite...

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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 or nite ordinals and cardinals, the distinction is linguistic and conceptual, not mathemat-ical. Transnite 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.) Transnite 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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