Combining Texts

All the ideas for 'Infinity: Quest to Think the Unthinkable', 'The Problem of Natural Laws' and 'Essays on Intellectual Powers 2: Senses'

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31 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 [Clegg]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10857 Set theory made a closer study of infinity possible [Clegg]
10864 Any set can always generate a larger set - its powerset, of subsets [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
10860 An ordinal number is defined by the set that comes before it [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10854 Transcendental numbers can't be fitted to finite equations [Clegg]
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 [Clegg]
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 [Clegg]
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 [Clegg]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
7. Existence / A. Nature of Existence / 1. Nature of Existence
23634 Accepting the existence of anything presupposes the notion of existence [Reid]
12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence
23635 Truths are self-evident to sensible persons who understand them clearly without prejudice [Reid]
12. Knowledge Sources / B. Perception / 1. Perception
7631 Sensation is not committed to any external object, but perception is [Reid]
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / c. Primary qualities
23637 Primary qualities are the object of mathematics [Reid]
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / d. Secondary qualities
23638 Secondary qualities conjure up, and are confused with, the sensations which produce them [Reid]
12. Knowledge Sources / B. Perception / 5. Interpretation
23639 It is unclear whether a toothache is in the mind or in the tooth, but the word has a single meaning [Reid]
12. Knowledge Sources / E. Direct Knowledge / 1. Common Sense
6492 Reid is seen as the main direct realist of the eighteenth century [Reid, by Robinson,H]
13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / a. Evidence
23641 People dislike believing without evidence, and try to avoid it [Reid]
13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / b. Pro-coherentism
23642 If non-rational evidence reaches us, it is reason which then makes use of it [Reid]
18. Thought / E. Abstraction / 2. Abstracta by Selection
23640 Only mature minds can distinguish the qualities of a body [Reid]
26. Natural Theory / D. Laws of Nature / 3. Laws and Generalities
4800 Natural laws result from eliminative induction, where enumerative induction gives generalisations [Cohen,LJ, by Psillos]