Combining Texts

All the ideas for 'Why coherence is not enough', 'Identity and Essence' and 'Infinity: Quest to Think the Unthinkable'

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35 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
10860 An ordinal number is defined by the set that comes before it [Clegg]
10861 Beyond infinity cardinals and ordinals can come apart [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]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
12132 Indiscernibility is a necessary and sufficient condition for identity [Brody]
9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
15834 Brody bases sortal essentialism on properties required throughout something's existence [Brody, by Mackie,P]
9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism
12140 Modern emphasis is on properties had essentially; traditional emphasis is on sort-defining properties [Brody]
9. Objects / D. Essence of Objects / 5. Essence as Kind
11895 A sortal essence is a property which once possessed always possessed [Brody, by Mackie,P]
12141 Maybe essential properties are those which determine a natural kind? [Brody]
9. Objects / D. Essence of Objects / 6. Essence as Unifier
12137 De re essentialism standardly says all possible objects identical with a have a's essential properties [Brody]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
12142 Essentially, a has P, always had P, must have had P, and has never had a future without P [Brody]
12143 An object having a property essentially is equivalent to its having it necessarily [Brody]
9. Objects / D. Essence of Objects / 8. Essence as Explanatory
12144 Essentialism is justified if the essential properties of things explain their other properties [Brody]
9. Objects / D. Essence of Objects / 12. Essential Parts
12139 Mereological essentialism says that every part that ensures the existence is essential [Brody]
9. Objects / E. Objects over Time / 12. Origin as Essential
12135 Interrupted objects have two first moments of existence, which could be two beginnings [Brody]
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
12130 a and b share all properties; so they share being-identical-with-a; so a = b [Brody]
10. Modality / E. Possible worlds / 3. Transworld Objects / b. Rigid designation
12138 Identity across possible worlds is prior to rigid designation [Brody]
13. Knowledge Criteria / A. Justification Problems / 2. Justification Challenges / a. Agrippa's trilemma
8840 There are five possible responses to the problem of infinite regress in justification [Cleve]
13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / a. Foundationalism
8841 Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve]