Ideas of Brian Clegg, by Theme
[British, fl. 2003, Technical consultant and freelance author.]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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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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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
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10857
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Set theory made a closer study of infinity possible
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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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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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
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10871
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Axiom of Existence: there exists at least one set
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
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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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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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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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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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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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10861
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Beyond infinity cardinals and ordinals can come apart
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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10854
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Transcendental numbers can't be fitted to finite equations
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
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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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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
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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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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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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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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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10869
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The Continuum Hypothesis is independent of the axioms of set theory
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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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