history-page8 - Infnite Ordinals and Cardinals • ²or...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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?
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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.)...
View Full Document

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.

Ask a homework question - tutors are online