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Single Idea 11042
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
]
Full Idea
A line is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point.
Gist of Idea
Parts of a line join at a point, so it is continuous
Source
Aristotle (Categories [c.331 BCE], 04b33)
Book Ref
Aristotle: 'Categories and De Interpretatione', ed/tr. Ackrill,J.R. [OUP 1963], p.13
A Reaction
This appears to be the essential concept of a Dedekind cut. It seems to be an open question whether a cut defines a unique number, but a boundary seems to be intrinsically unique. Aristotle wins again.
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]
|
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]
|