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
10859

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

4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10857

Set theory made a closer study of infinity possible

10864

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
10872

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
10875

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
10876

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
10878

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
10877

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10879

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
10871

Axiom of Existence: there exists at least one set

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
10874

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

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10880

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10860

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

10861

Beyond infinity cardinals and ordinals can come apart

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10854

Transcendental numbers can't be fitted to finite equations

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
10858

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
10853

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
10866

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10869

The Continuum Hypothesis is independent of the axioms of set theory

10862

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