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

[all numbers, including those inexpressible as fractions]

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