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] |