44 ideas
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [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] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [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] |
| 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] |
| 21382 | Things get smaller without end [Anaxagoras] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 481 | Nothing is created or destroyed; there is only mixing and separation [Anaxagoras] |
| 21822 | Anaxagoras's concept of supreme Mind has a simple First and a multiple One [Anaxagoras, by Plotinus] |
| 17995 | Basic is the potentially perceptible, then comes the contrary qualities, and finally the 'elements' [Anaxagoras] |
| 20802 | Snow is not white, and doesn't even appear white, because it is made of black water [Anaxagoras, by Cicero] |
| 13257 | The senses are too feeble to determine the truth [Anaxagoras] |
| 22761 | We reveal unreliability in the senses when we cannot discriminate a slow change of colour [Anaxagoras, by Sext.Empiricus] |
| 13256 | Nous is unlimited, self-ruling and pure; it is the finest thing, with great discernment and strength [Anaxagoras] |
| 13784 | Mind is self-ruling, pure, ordering and ubiquitous [Anaxagoras, by Plato] |
| 5118 | Anaxagoras says mind remains pure, and so is not affected by what it changes [Anaxagoras, by Aristotle] |
| 20328 | A thing is only seen as art in an 'artworld', which has a theory and a history [Danto] |
| 20441 | An ordinary object can be a work of art, but only if some theory of art supports it [Danto] |
| 18231 | Anaxagoras said a person would choose to be born to contemplate the ordered heavens [Anaxagoras] |
| 631 | For Anaxagoras the Good Mind has no opposite, and causes all movement, for a higher reason [Anaxagoras, by Aristotle] |
| 22727 | Mind creates the world from a mixture of pure substances [Anaxagoras, by ] |
| 550 | Anaxagoras said that the number of principles was infinite [Anaxagoras, by Aristotle] |
| 21383 | The ultimate constituents of reality are the homoeomeries [Anaxagoras, by Vlastos] |
| 13208 | Anaxagoreans regard the homoeomeries as elements, which compose earth, air, fire and water [Anaxagoras, by Aristotle] |
| 367 | Anaxagoras says mind produces order and causes everything [Anaxagoras, by Plato] |
| 21381 | Germs contain microscopic organs, which become visible as they grow [Anaxagoras] |
| 22726 | When things were unified, Mind set them in order [Anaxagoras] |
| 2629 | Anaxagoras was the first to say that the universe is directed by an intelligence [Anaxagoras, by Cicero] |
| 480 | Past, present and future, and the movements of the heavens, were arranged by Mind [Anaxagoras] |
| 7488 | Anaxagoras was the first recorded atheist [Anaxagoras, by Watson] |
| 5956 | Anaxagoras was charged with impiety for calling the sun a lump of stone [Anaxagoras, by Plutarch] |