Ideas from 'Infinity: Quest to Think the Unthinkable' by Brian Clegg [2003], by Theme Structure
[found in 'Infinity' by Clegg,Brian [Robinson 2003,978-1-84119-650-3]].
green numbers give full details |
back to texts
|
expand these ideas
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
|
10859
|
A set is 'well-ordered' 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 non-empty 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 aleph-one is the cardinality of the reals
|