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Ideas of Brian Clegg, by Text
[British, fl. 2003, Technical consultant and freelance author.]
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2003
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Infinity: Quest to Think the Unthinkable
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Ch. 6
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p.61
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10853
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Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless
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Ch. 6
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p.69
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10854
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Transcendental numbers can't be fitted to finite equations
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Ch.12
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p.163
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10858
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By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line
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Ch.13
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p.157
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10857
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Set theory made a closer study of infinity possible
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Ch.13
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p.168
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10859
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A set is 'well-ordered' if every subset has a first element
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Ch.13
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p.169
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10860
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An ordinal number is defined by the set that comes before it
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Ch.13
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p.169
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10861
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Beyond infinity cardinals and ordinals can come apart
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Ch.14
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p.179
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10862
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The 'continuum hypothesis' says aleph-one is the cardinality of the reals
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Ch.14
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p.184
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10864
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Any set can always generate a larger set - its powerset, of subsets
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Ch.15
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p.193
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10866
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Cantor's account of infinities has the shaky foundation of irrational numbers
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Ch.15
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p.204
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10869
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The Continuum Hypothesis is independent of the axioms of set theory
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Ch.15
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p.205
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10875
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Pairing: For any two sets there exists a set to which they both belong
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Ch.15
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p.205
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10871
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Axiom of Existence: there exists at least one set
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Ch.15
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p.205
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10874
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Specification: a condition applied to a set will always produce a new set
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Ch.15
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p.205
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10872
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Extensionality: Two sets are equal if and only if they have the same elements
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Ch.15
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p.206
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10878
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Infinity: There exists a set of the empty set and the successor of each element
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Ch.15
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p.206
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10879
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Choice: For every set a mechanism will choose one member of any non-empty subset
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Ch.15
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p.206
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10876
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Unions: There is a set of all the elements which belong to at least one set in a collection
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Ch.15
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p.206
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10877
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Powers: All the subsets of a given set form their own new powerset
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Ch.17
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p.218
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10880
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Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable)
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