Combining Texts

All the ideas for 'Philosophical Explanations', 'Intro to 'Modality and Tense'' and 'Investigations in the Foundations of Set Theory I'

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25 ideas

1. Philosophy / D. Nature of Philosophy / 7. Despair over Philosophy
9208 Philosophers with a new concept are like children with a new toy [Fine,K]
2. Reason / D. Definition / 8. Impredicative Definition
15924 Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
17608 We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
17607 Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
13012 Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
17609 Set theory can be reduced to a few definitions and seven independent axioms [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
13015 Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
13020 The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
13486 Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13487 In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13027 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
9627 Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
9210 Possible objects are abstract; actual concrete objects are possible; so abstract/concrete are compatible [Fine,K]
7. Existence / D. Theories of Reality / 3. Reality
9211 A non-standard realism, with no privileged standpoint, might challenge its absoluteness or coherence [Fine,K]
9. Objects / A. Existence of Objects / 3. Objects in Thought
9202 Objects, as well as sentences, can have logical form [Fine,K]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / b. Essence not necessities
9206 We must distinguish between the identity or essence of an object, and its necessary features [Fine,K]
10. Modality / A. Necessity / 3. Types of Necessity
9205 The three basic types of necessity are metaphysical, natural and normative [Fine,K]
10. Modality / A. Necessity / 5. Metaphysical Necessity
9209 Metaphysical necessity may be 'whatever the circumstance', or 'regardless of circumstances' [Fine,K]
10. Modality / A. Necessity / 11. Denial of Necessity
9200 Empiricists suspect modal notions: either it happens or it doesn't; it is just regularities. [Fine,K]
11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs
3570 Maybe knowledge is belief which 'tracks' the truth [Nozick, by Williams,M]
13. Knowledge Criteria / C. External Justification / 4. Tracking the Facts
2748 A true belief isn't knowledge if it would be believed even if false. It should 'track the truth' [Nozick, by Dancy,J]
19. Language / C. Assigning Meanings / 8. Possible Worlds Semantics
9207 If sentence content is all worlds where it is true, all necessary truths have the same content! [Fine,K]