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All the ideas for 'Investigations in the Foundations of Set Theory I', 'works' and 'Causal and Metaphysical Necessity'

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29 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 / 2. History of Logic
11022 Gentzen introduced a natural deduction calculus (NK) in 1934 [Gentzen, by Read]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
11065 The inferential role of a logical constant constitutes its meaning [Gentzen, by Hanna]
11023 The logical connectives are 'defined' by their introduction rules [Gentzen]
11213 Each logical symbol has an 'introduction' rule to define it, and hence an 'elimination' rule [Gentzen]
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
15091 Restrict 'logical truth' to formal logic, rather than including analytic and metaphysical truths [Shoemaker]
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 / 4. Axioms for Number / g. Incompleteness of Arithmetic
10067 Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave]
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]
8. Modes of Existence / B. Properties / 1. Nature of Properties
15095 A property's causal features are essential, and only they fix its identity [Shoemaker]
15097 I claim that a property has its causal features in all possible worlds [Shoemaker]
8. Modes of Existence / C. Powers and Dispositions / 3. Powers as Derived
15094 I now deny that properties are cluster of powers, and take causal properties as basic [Shoemaker]
10. Modality / A. Necessity / 5. Metaphysical Necessity
15099 If something is possible, but not nomologically possible, we need metaphysical possibility [Shoemaker]
10. Modality / D. Knowledge of Modality / 1. A Priori Necessary
15101 Once you give up necessity as a priori, causal necessity becomes the main type of necessity [Shoemaker]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / a. Conceivable as possible
15098 Empirical evidence shows that imagining a phenomenon can show it is possible [Shoemaker]
15100 Imagination reveals conceptual possibility, where descriptions avoid contradiction or incoherence [Shoemaker]
14. Science / C. Induction / 5. Paradoxes of Induction / a. Grue problem
15096 'Grue' only has causal features because of its relation to green [Shoemaker]
26. Natural Theory / D. Laws of Nature / 5. Laws from Universals
15093 We might say laws are necessary by combining causal properties with Armstrong-Dretske-Tooley laws [Shoemaker]