31 ideas
8368 | A correct definition is what can be substituted without loss of meaning [Ducasse] |
10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
10857 | Set theory made a closer study of infinity possible [Clegg] |
10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
10871 | Axiom of Existence: there exists at least one set [Clegg] |
10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
10860 | An ordinal number is defined by the set that comes before it [Clegg] |
10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
9073 | Abstraction from an ambiguous concept like 'mole' will define them as the same [Barnes,J] |
9074 | Abstraction cannot produce the concept of a 'game', as there is no one common feature [Barnes,J] |
9072 | Defining concepts by abstractions will collect together far too many attributes from entities [Barnes,J] |
8367 | Causation is defined in terms of a single sequence, and constant conjunction is no part of it [Ducasse] |
8372 | We see what is in common between causes to assign names to them, not to perceive them [Ducasse] |
8369 | Causes are either sufficient, or necessary, or necessitated, or contingent upon [Ducasse] |
8373 | When a brick and a canary-song hit a window, we ignore the canary if we are interested in the breakage [Ducasse] |
8370 | A cause is a change which occurs close to the effect and just before it [Ducasse] |
8371 | Recurrence is only relevant to the meaning of law, not to the meaning of cause [Ducasse] |
8374 | We are interested in generalising about causes and effects purely for practical purposes [Ducasse] |