Combining Philosophers

All the ideas for R Feldman / E Conee, Ernst Zermelo and Bob Hale/ Aviv Hoffmann

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26 ideas

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]
9565 Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara]
3339 For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn]
17832 Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M]
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 / h. Axiom of Replacement VII
13028 Replacement was added when some advanced theorems seemed to need it [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]
5. Theory of Logic / L. Paradox / 3. Antinomies
17626 The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo]
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 / A. Nature of Mathematics / 5. The Infinite / e. Countable infinity
15897 Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine]
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]
10. Modality / A. Necessity / 1. Types of Modality
14528 Maybe modal thought is unavoidable, as a priori recognition of necessary truth-preservation in reasoning [Hale/Hoffmann,A]
11. Knowledge Aims / A. Knowledge / 4. Belief / a. Beliefs
8808 Involuntary beliefs can still be evaluated [Feldman/Conee]
13. Knowledge Criteria / B. Internal Justification / 3. Evidentialism / b. Evidentialism
8807 Evidentialism is the view that justification is determined by the quality of the evidence [Feldman/Conee]
8809 Beliefs should fit evidence, and if you ought to believe it, then you are justified [Feldman/Conee]
13. Knowledge Criteria / C. External Justification / 3. Reliabilism / a. Reliable knowledge
8810 If someone rejects good criticism through arrogance, that is irrelevant to whether they have knowledge [Feldman/Conee]
18. Thought / A. Modes of Thought / 6. Judgement / a. Nature of Judgement
17613 We should judge principles by the science, not science by some fixed principles [Zermelo]