60 ideas
3750 | "It is true that x" means no more than x [Ramsey] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
13430 | Infinity: there is an infinity of distinguishable individuals [Ramsey] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
13428 | Reducibility: to every non-elementary function there is an equivalent elementary function [Ramsey] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
13427 | Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey] |
13334 | Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
6409 | The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling] |
13426 | Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey] |
13425 | Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
16659 | Relations do not add anything to reality, though they are real aspects of the world [Olivi] |
8495 | The distinction between particulars and universals is a mistake made because of language [Ramsey] |
8493 | We could make universals collections of particulars, or particulars collections of their qualities [Ramsey] |
8494 | Obviously 'Socrates is wise' and 'Socrates has wisdom' express the same fact [Ramsey] |
16672 | Quantity is the quantified parts of a thing, plus location and coordination [Olivi] |
16673 | Quantity just adds union and location to the extension of parts [Olivi] |
13766 | 'If' is the same as 'given that', so the degrees of belief should conform to probability theory [Ramsey, by Ramsey] |
10993 | Ramsey's Test: believe the consequent if you believe the antecedent [Ramsey, by Read] |
14279 | Asking 'If p, will q?' when p is uncertain, then first add p hypothetically to your knowledge [Ramsey] |
3212 | Beliefs are maps by which we steer [Ramsey] |
22328 | I just confront the evidence, and let it act on me [Ramsey] |
22325 | A belief is knowledge if it is true, certain and obtained by a reliable process [Ramsey] |
19724 | Belief is knowledge if it is true, certain, and obtained by a reliable process [Ramsey] |
6894 | Mental terms can be replaced in a sentence by a variable and an existential quantifier [Ramsey] |
19143 | Ramsey gave axioms for an uncertain agent to decide their preferences [Ramsey, by Davidson] |
18818 | Sentence meaning is given by the actions to which it would lead [Ramsey] |
9418 | All knowledge needs systematizing, and the axioms would be the laws of nature [Ramsey] |
9420 | Causal laws result from the simplest axioms of a complete deductive system [Ramsey] |
16663 | Things are limited by the species to certain modes of being [Olivi] |