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### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

#### [all numbers, including those inexpressible as fractions]

29 ideas
 11042 Parts of a line join at a point, so it is continuous [Aristotle]
 22962 Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle]
 13445 Descartes showed a one-one order-preserving match between points on a line and the real numbers [Descartes, by Hart,WD]
 17611 We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
 11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
 15906 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
 18252 Real numbers are ratios of quantities, such as lengths or masses [Frege]
 9889 Real numbers are ratios of quantities [Frege, by Dummett]
 18253 I wish to go straight from cardinals to reals (as ratios), leaving out the rationals [Frege]
 14135 Real numbers are a class of rational numbers (and so not really numbers at all) [Russell]
 18738 We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein]
 14648 Could I name all of the real numbers in one fell swoop? Call them all 'Charley'? [Plantinga]
 18095 Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
 18099 The number of reals is the number of subsets of the natural numbers [Bostock]
 10610 The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P]
 12395 Real numbers stand to measurement as natural numbers stand to counting [Kitcher]
 13446 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
 17784 Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
 10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
 18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
 10165 'Analysis' is the theory of the real numbers [Reck/Price]
 10632 The real numbers may be introduced by abstraction as ratios of quantities [Hale, by Hale/Wright]
 10107 Real numbers provide answers to square root problems [George/Velleman]
 15711 The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan]
 10854 Transcendental numbers can't be fitted to finite equations [Clegg]
 15922 For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
 8671 The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
 10684 I take the real numbers to be just lengths [Hossack]
 15364 English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten]