Combining Texts

All the ideas for 'Philosophical Fragments', 'Carnap and Logical Truth' and 'Investigations in the Foundations of Set Theory I'

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25 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]
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]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
13010 In order to select the logic justified by experience, we would need to use a lot of logic [Boghossian on Quine]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
9002 Elementary logic requires truth-functions, quantifiers (and variables), identity, and also sets of variables [Quine]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
13681 Logical consequence is marked by being preserved under all nonlogical substitutions [Quine, by Sider]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13829 If logical truths essentially depend on logical constants, we had better define the latter [Hacking on Quine]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
9003 Set theory was struggling with higher infinities, when new paradoxes made it baffling [Quine]
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]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9004 If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic [Quine]
7. Existence / A. Nature of Existence / 5. Reason for Existence
16007 I assume existence, rather than reasoning towards it [Kierkegaard]
8. Modes of Existence / E. Nominalism / 1. Nominalism / b. Nominalism about universals
9006 Commitment to universals is as arbitrary or pragmatic as the adoption of a new system of bookkeeping [Quine]
10. Modality / A. Necessity / 2. Nature of Necessity
16013 Nothing necessary can come into existence, since it already 'is' [Kierkegaard]
10. Modality / A. Necessity / 6. Logical Necessity
9001 Frege moved Kant's question about a priori synthetic to 'how is logical certainty possible?' [Quine]
12. Knowledge Sources / A. A Priori Knowledge / 7. A Priori from Convention
9005 Examination of convention in the a priori begins to blur the distinction with empirical knowledge [Quine]