Combining Texts

All the ideas for 'Vagaries of Definition', 'A Completeness Theorem in Modal Logic' and 'What Numbers Could Not Be'

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31 ideas

4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
10163 Propositional modal logic has been proved to be complete [Kripke, by Feferman/Feferman]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
10760 With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound [Kripke, by Rossberg]
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
16189 The variable domain approach to quantified modal logic invalidates the Barcan Formula [Kripke, by Simchen]
15132 The Barcan formulas fail in models with varying domains [Kripke, by Williamson]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9912 There are no such things as numbers [Benacerraf]
9901 Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9151 Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
13891 To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C]
17904 A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf]
17906 To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
9898 We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf]
17903 Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
9897 The application of a system of numbers is counting and measurement [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
8203 All the arithmetical entities can be reduced to classes of integers, and hence to sets [Quine]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
8697 Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
8304 No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
9906 If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
9908 The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
9909 The number 3 defines the role of being third in a progression [Benacerraf]
9911 Number words no more have referents than do the parts of a ruler [Benacerraf]
8925 Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
9938 How can numbers be objects if order is their only property? [Benacerraf, by Putnam]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
9910 Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9903 Number words are not predicates, as they function very differently from adjectives [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9904 The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
9. Objects / F. Identity among Objects / 6. Identity between Objects
9905 Identity statements make sense only if there are possible individuating conditions [Benacerraf]
19. Language / A. Nature of Meaning / 10. Denial of Meanings
8202 Meaning is essence divorced from things and wedded to words [Quine]
19. Language / E. Analyticity / 4. Analytic/Synthetic Critique
8201 The distinction between meaning and further information is as vague as the essence/accident distinction [Quine]