Combining Texts

All the ideas for 'works', 'Meinong on Complexes and Assumptions' and 'New system of communication of substances'

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60 ideas

3. Truth / B. Truthmakers / 6. Making Negative Truths
21544 It seems that when a proposition is false, something must fail to subsist [Russell]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
15901 Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
13444 Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
18098 Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
15505 If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10865 The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
10701 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
13016 The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
14199 Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
21539 Excluded middle can be stated psychologically, as denial of p implies assertion of not-p [Russell]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
10082 There are infinite sets that are not enumerable [Cantor, by Smith,P]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
13483 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
8710 The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
15910 Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
15905 Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
17798 Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
14136 A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
15906 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
15903 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
18251 Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
15902 Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
15908 It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
13464 Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
10112 The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17889 CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
13447 Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
10883 Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
13528 Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
9555 Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
18174 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18173 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
10232 Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
18176 Pure mathematics is pure set theory [Cantor]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
8631 Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
7. Existence / D. Theories of Reality / 2. Realism
21538 If two people perceive the same object, the object of perception can't be in the mind [Russell]
8. Modes of Existence / A. Relations / 1. Nature of Relations
21534 The only thing we can say about relations is that they relate [Russell]
21540 Relational propositions seem to be 'about' their terms, rather than about the relation [Russell]
8. Modes of Existence / C. Powers and Dispositions / 4. Powers as Essence
13169 I call Aristotle's entelechies 'primitive forces', which originate activity [Leibniz]
13168 My formal unifying atoms are substantial forms, which are forces like appetites [Leibniz]
9. Objects / A. Existence of Objects / 3. Objects in Thought
21536 When I perceive a melody, I do not perceive the notes as existing [Russell]
9. Objects / A. Existence of Objects / 5. Simples
13170 The analysis of things leads to atoms of substance, which found both composition and action [Leibniz]
9. Objects / A. Existence of Objects / 5. Individuation / c. Individuation by location
21535 Objects only exist if they 'occupy' space and time [Russell]
9. Objects / B. Unity of Objects / 2. Substance / c. Types of substance
13171 Substance must necessarily involve progress and change [Leibniz]
10. Modality / B. Possibility / 5. Contingency
21533 Contingency arises from tensed verbs changing the propositions to which they refer [Russell]
11. Knowledge Aims / C. Knowing Reality / 1. Perceptual Realism / b. Direct realism
21537 I assume we perceive the actual objects, and not their 'presentations' [Russell]
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
21532 Full empiricism is not tenable, but empirical investigation is always essential [Russell]
18. Thought / A. Modes of Thought / 6. Judgement / b. Error
21542 Do incorrect judgements have non-existent, or mental, or external objects? [Russell]
18. Thought / C. Content / 1. Content
21541 The complexity of the content correlates with the complexity of the object [Russell]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
8715 Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
18. Thought / E. Abstraction / 2. Abstracta by Selection
13454 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
19. Language / D. Propositions / 1. Propositions
21543 If p is false, then believing not-p is knowing a truth, so negative propositions must exist [Russell]
27. Natural Reality / A. Classical Physics / 1. Mechanics / c. Forces
13167 We need the metaphysical notion of force to explain mechanics, and not just extended mass [Leibniz]
27. Natural Reality / C. Space / 3. Points in Space
10863 Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
28. God / A. Divine Nature / 2. Divine Nature
13465 Only God is absolutely infinite [Cantor, by Hart,WD]