Combining Philosophers

Ideas for Rescher,N/Oppenheim,P, Bertrand Russell and David Lewis

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44 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
18395 Sets are mereological sums of the singletons of their members [Lewis, by Armstrong]
15496 We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton [Lewis]
10807 Mathematics reduces to set theory, which reduces, with some mereology, to the singleton function [Lewis]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
15500 Classes divide into subclasses in many ways, but into members in only one way [Lewis]
15499 A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
14113 The null class is a fiction [Russell]
15503 We needn't accept this speck of nothingness, this black hole in the fabric of Reality! [Lewis]
15498 We can accept the null set, but there is no null class of anything [Lewis]
15502 There are four main reasons for asserting that there is an empty set [Lewis]
10809 We can accept the null set, but not a null class, a class lacking members [Lewis]
10811 The null set plays the role of last resort, for class abstracts and for existence [Lewis]
10812 The null set is not a little speck of sheer nothingness, a black hole in Reality [Lewis]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
15506 If we don't understand the singleton, then we don't understand classes [Lewis]
6103 Normally a class with only one member is a problem, because the class and the member are identical [Russell]
15497 We can replace the membership relation with the member-singleton relation (plus mereology) [Lewis]
15511 If singleton membership is external, why is an object a member of one rather than another? [Lewis]
15513 Maybe singletons have a structure, of a thing and a lasso? [Lewis]
10813 What on earth is the relationship between a singleton and an element? [Lewis]
10814 Are all singletons exact intrinsic duplicates? [Lewis]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
14427 We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
15507 Set theory has some unofficial axioms, generalisations about how to understand it [Lewis]
10191 Set theory reduces to a mereological theory with singletons as the only atoms [Lewis, by MacBride]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
14428 Members define a unique class, whereas defining characteristics are numerous [Russell]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
14440 We may assume that there are infinite collections, as there is no logical reason against them [Russell]
14447 Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
14443 The British parliament has one representative selected from each constituency [Russell]
14445 Choice shows that if any two cardinals are not equal, one must be the greater [Russell]
14444 Choice is equivalent to the proposition that every class is well-ordered [Russell]
14446 We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
18130 Axiom of Reducibility: there is always a function of the lowest possible order in a given level [Russell, by Bostock]
14459 Reducibility: a family of functions is equivalent to a single type of function [Russell]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
15508 If singletons are where their members are, then so are all sets [Lewis]
15514 A huge part of Reality is only accepted as existing if you have accepted set theory [Lewis]
15523 Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it [Lewis]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
21563 The 'no classes' theory says the propositions just refer to the members [Russell]
14461 Propositions about classes can be reduced to propositions about their defining functions [Russell]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
15894 Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
14126 Order rests on 'between' and 'separation' [Russell]
14127 Order depends on transitive asymmetrical relations [Russell]
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
8469 Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
11064 Classes can be reduced to propositional functions [Russell, by Hanna]
7548 Classes, grouped by a convenient property, are logical constructions [Russell]
8745 Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell]
6436 I gradually replaced classes with properties, and they ended as a symbolic convenience [Russell]