Combining Texts

All the ideas for 'works', 'Culture and Value' and 'Letter to Menoeceus'

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59 ideas

1. Philosophy / A. Wisdom / 2. Wise People
16010 While faith is a passion (as Kierkegaard says), wisdom is passionless [Wittgenstein]
1. Philosophy / D. Nature of Philosophy / 2. Invocation to Philosophy
14052 Begin philosophy when you are young, and keep going when you are old [Epicurus]
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
10701 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
10865 The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
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 / 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
8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
17889 CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
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
18174 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
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]
16. Persons / F. Free Will / 6. Determinism / b. Fate
14062 Sooner follow mythology, than accept the 'fate' of natural philosophers [Epicurus]
16. Persons / F. Free Will / 7. Compatibilism
1837 We should not refer things to irresponsible necessity, but either to fortune or to our own will [Epicurus]
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]
20. Action / C. Motives for Action / 3. Acting on Reason / a. Practical reason
1836 Prudence is more valuable than philosophy, because it avoids confusions of the soul [Epicurus]
20. Action / C. Motives for Action / 4. Responsibility for Actions
14061 Our own choices are autonomous, and the basis for praise and blame [Epicurus]
22. Metaethics / B. Value / 2. Values / e. Death
14054 Fearing death is absurd, because we are not present when it occurs [Epicurus]
14053 It is absurd to fear the pain of death when you are not even facing it [Epicurus]
14055 The wisdom that produces a good life also produces a good death [Epicurus]
22. Metaethics / C. The Good / 1. Goodness / f. Good as pleasure
14057 All pleasures are good, but it is not always right to choose them [Epicurus]
14058 Pleasure is the goal, but as lack of pain and calm mind, not as depraved or greedy pleasure [Epicurus]
1833 Pleasure is the first good in life [Epicurus]
22. Metaethics / C. The Good / 1. Goodness / i. Moral luck
14063 Sooner a good decision going wrong, than a bad one turning out for the good [Epicurus]
22. Metaethics / C. The Good / 2. Happiness / d. Routes to happiness
14059 The best life is not sensuality, but rational choice and healthy opinion [Epicurus]
22. Metaethics / C. The Good / 3. Pleasure / a. Nature of pleasure
1835 True pleasure is not debauchery, but freedom from physical and mental pain [Epicurus]
22. Metaethics / C. The Good / 3. Pleasure / c. Value of pleasure
14056 We only need pleasure when we have the pain of desire [Epicurus]
23. Ethics / C. Virtue Theory / 1. Virtue Theory / b. Basis of virtue
14060 Prudence is the greatest good, and more valuable than philosophy, because it produces virtue [Epicurus]
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]