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Single Idea 10610

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers ]

Full Idea

It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers.

Gist of Idea

The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals

Source

Peter Smith (Intro to Gödel's Theorems [2007], 18.2)

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.157

The 29 ideas with the same theme [all numbers, including those inexpressible as fractions]:

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]
15906 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
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]