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Ideas of Brian Clegg, by Text
[British, fl. 2003, Technical consultant and freelance author.]
2003

Infinity: Quest to Think the Unthinkable

Ch. 6

p.61

10853

Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless

Ch. 6

p.69

10854

Transcendental numbers can't be fitted to finite equations

Ch.12

p.163

10858

By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line

Ch.13

p.157

10857

Set theory made a closer study of infinity possible

Ch.13

p.168

10859

A set is 'wellordered' if every subset has a first element

Ch.13

p.169

10861

Beyond infinity cardinals and ordinals can come apart

Ch.13

p.169

10860

An ordinal number is defined by the set that comes before it

Ch.14

p.179

10862

The 'continuum hypothesis' says alephone is the cardinality of the reals

Ch.14

p.184

10864

Any set can always generate a larger set  its powerset, of subsets

Ch.15

p.193

10866

Cantor's account of infinities has the shaky foundation of irrational numbers

Ch.15

p.204

10869

The Continuum Hypothesis is independent of the axioms of set theory

Ch.15

p.205

10875

Pairing: For any two sets there exists a set to which they both belong

Ch.15

p.205

10874

Specification: a condition applied to a set will always produce a new set

Ch.15

p.205

10871

Axiom of Existence: there exists at least one set

Ch.15

p.205

10872

Extensionality: Two sets are equal if and only if they have the same elements

Ch.15

p.206

10876

Unions: There is a set of all the elements which belong to at least one set in a collection

Ch.15

p.206

10878

Infinity: There exists a set of the empty set and the successor of each element

Ch.15

p.206

10879

Choice: For every set a mechanism will choose one member of any nonempty subset

Ch.15

p.206

10877

Powers: All the subsets of a given set form their own new powerset

Ch.17

p.218

10880

Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable)
