Combining Texts

All the ideas for 'works', 'Fourfold Root of Princ of Sufficient Reason' and 'Elements of Set Theory'

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17 ideas

2. Reason / B. Laws of Thought / 2. Sufficient Reason
21918 Sufficient Reason can't be proved, because all proof presupposes it [Schopenhauer, by Lewis,PB]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
13204 The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
13206 A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13200 Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]
13199 The empty set may look pointless, but many sets can be constructed from it [Enderton]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
13203 The singleton is defined using the pairing axiom (as {x,x}) [Enderton]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13202 Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13205 We can only define functions if Choice tells us which items are involved [Enderton]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13007 Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
7. Existence / E. Categories / 1. Categories
21920 No need for a priori categories, since sufficient reason shows the interrelations [Schopenhauer, by Lewis,PB]
10. Modality / A. Necessity / 3. Types of Necessity
21362 Necessity is physical, logical, mathematical or moral [Schopenhauer, by Janaway]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / a. Idealism
21361 For Schopenhauer, material things would not exist without the mind [Schopenhauer, by Janaway]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
21919 Object for a subject and representation are the same thing [Schopenhauer]
14. Science / D. Explanation / 2. Types of Explanation / a. Types of explanation
21917 The four explanations: objects by causes, concepts by ground, maths by spacetime, ethics by motive [Schopenhauer, by Lewis,PB]
18. Thought / D. Concepts / 2. Origin of Concepts / b. Empirical concepts
21921 Concepts are abstracted from perceptions [Schopenhauer, by Lewis,PB]
20. Action / C. Motives for Action / 3. Acting on Reason / c. Reasons as causes
21363 Motivation is causality seen from within [Schopenhauer]