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Single Idea 14436
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
]
Full Idea
There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'.
Gist of Idea
A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.69
Related Idea
Idea 14437
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
The
54 ideas
from 'Introduction to Mathematical Philosophy'
8468
|
The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false
[Russell, by Orenstein]
|
8469
|
Russell's proposal was that only meaningful predicates have sets as their extensions
[Russell, by Orenstein]
|
13414
|
For Russell, numbers are sets of equivalent sets
[Russell, by Benacerraf]
|
14420
|
Infinity and continuity used to be philosophy, but are now mathematics
[Russell]
|
8745
|
Classes are logical fictions, and are not part of the ultimate furniture of the world
[Russell]
|
14421
|
Discovering that 1 is a number was difficult
[Russell]
|
14424
|
Numbers are needed for counting, so they need a meaning, and not just formal properties
[Russell]
|
14423
|
'0', 'number' and 'successor' cannot be defined by Peano's axioms
[Russell]
|
14422
|
Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms
[Russell]
|
14425
|
A number is something which characterises collections of the same size
[Russell]
|
14427
|
We can enumerate finite classes, but an intensional definition is needed for infinite classes
[Russell]
|
14428
|
Members define a unique class, whereas defining characteristics are numerous
[Russell]
|
14426
|
A definition by 'extension' enumerates items, and one by 'intension' gives a defining property
[Russell]
|
14430
|
If a relation is symmetrical and transitive, it has to be reflexive
[Russell]
|
14429
|
Classes are logical fictions, made from defining characteristics
[Russell]
|
14431
|
The definition of order needs a transitive relation, to leap over infinite intermediate terms
[Russell]
|
14441
|
The formal laws of arithmetic are the Commutative, the Associative and the Distributive
[Russell]
|
10450
|
Russell admitted that even names could also be used as descriptions
[Russell, by Bach]
|
13510
|
Could a number just be something which occurs in a progression?
[Russell, by Hart,WD]
|
14432
|
'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a
[Russell]
|
14435
|
The essence of individuality is beyond description, and hence irrelevant to science
[Russell]
|
14433
|
Mathematically expressed propositions are true of the world, but how to interpret them?
[Russell]
|
14434
|
What matters is the logical interrelation of mathematical terms, not their intrinsic nature
[Russell]
|
14436
|
A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum
[Russell]
|
14439
|
A complex number is simply an ordered couple of real numbers
[Russell]
|
14438
|
New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
[Russell]
|
14440
|
We may assume that there are infinite collections, as there is no logical reason against them
[Russell]
|
14442
|
If straight lines were like ratios they might intersect at a 'gap', and have no point in common
[Russell]
|
14443
|
The British parliament has one representative selected from each constituency
[Russell]
|
14444
|
Choice is equivalent to the proposition that every class is well-ordered
[Russell]
|
14445
|
Choice shows that if any two cardinals are not equal, one must be the greater
[Russell]
|
14446
|
We can pick all the right or left boots, but socks need Choice to insure the representative class
[Russell]
|
14447
|
Infinity says 'for any inductive cardinal, there is a class having that many terms'
[Russell]
|
14449
|
There is always something psychological about inference
[Russell]
|
12197
|
Inferring q from p only needs p to be true, and 'not-p or q' to be true
[Russell]
|
14450
|
All forms of implication are expressible as truth-functions
[Russell]
|
14451
|
Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts
[Russell]
|
14454
|
An argument 'satisfies' a function φx if φa is true
[Russell]
|
14453
|
The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M?
[Russell]
|
14452
|
All the propositions of logic are completely general
[Russell]
|
12444
|
Logic is concerned with the real world just as truly as zoology
[Russell]
|
14456
|
'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity
[Russell]
|
14457
|
Names are really descriptions, except for a few words like 'this' and 'that'
[Russell]
|
14458
|
Asking 'Did Homer exist?' is employing an abbreviated description
[Russell]
|
7311
|
The only genuine proper names are 'this' and 'that'
[Russell]
|
14455
|
'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not
[Russell]
|
14459
|
Reducibility: a family of functions is equivalent to a single type of function
[Russell]
|
14461
|
Propositions about classes can be reduced to propositions about their defining functions
[Russell]
|
14460
|
If something is true in all possible worlds then it is logically necessary
[Russell]
|
14463
|
Existence can only be asserted of something described, not of something named
[Russell]
|
10057
|
Logic can only assert hypothetical existence
[Russell]
|
14464
|
Logic can be known a priori, without study of the actual world
[Russell]
|
14462
|
In modern times, logic has become mathematical, and mathematics has become logical
[Russell]
|
14465
|
Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men'
[Russell]
|