39 ideas
19695 | The devil was wise as an angel, and lost no knowledge when he rebelled [Whitcomb] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
23125 | Most good social changes are incremental, rather than revolutionary [Gopnik] |
23126 | Conservatives often want peace, prosperity and tolerance, but not social fairness [Gopnik] |
23132 | Conservatives believe obedience and rank are essential to social order [Gopnik] |
23142 | People are fallible, so liberalism tries to distribute power [Gopnik] |
23143 | Liberals have tried very hard to build a conscience into their institutions [Gopnik] |
23128 | The opposite of liberalism is dogmatism [Gopnik] |
23141 | Left-wingers are inconsistent in their essentialist descriptions of social groups [Gopnik] |
23124 | Liberal community is not blood ties or tradition, but shared choices, and sympathy for the losers [Gopnik] |
23127 | Liberal community includes flight from the family, into energetic reforming groups [Gopnik] |
23129 | Right-wingers attack liberal faith in reason, left-wingers attack its faith in reform [Gopnik] |
23133 | Cosmopolitan liberals lack national loyalty, and welcome excessive immigration [Gopnik] |
23138 | Modern left-wingers criticise liberalism's control of culture [Gopnik] |
23139 | Liberalism's attempt to be neutral and colour-blind erases cultural identities [Gopnik] |
23135 | Classic Marxists see liberalism as the ideology of the bourgeoisie [Gopnik] |
23140 | Environmental disasters result not from capitalism, but from a general drive for growth [Gopnik] |
23130 | Popular imperialism gives the poor the belief that their acts have world historical meaning [Gopnik] |
23131 | Patriots love their place, but nationalists have a paranoid ethnic hostility [Gopnik] |
23136 | Liberal free speech is actually paid speech [Gopnik] |
23134 | A 'free' society implies a free market, which always produces predatory capitalism and inequalities [Gopnik] |