35 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] |
| 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] |
| 20457 | Zeno assumes collecting an infinity of things makes an infinite thing [Rovelli] |
| 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] |
| 20468 | Quantum mechanics deals with processes, rather than with things [Rovelli] |
| 20467 | Quantum mechanics describes the world entirely as events [Rovelli] |
| 20469 | There are probably no infinities, and 'infinite' names what we do not yet know [Rovelli] |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| 20461 | The basic ideas of fields and particles are merged in quantum mechanics [Rovelli] |
| 20462 | Because it is quantised, a field behaves like a set of packets of energy [Rovelli] |
| 20463 | There are about fifteen particles fields, plus a few force fields [Rovelli] |
| 20464 | The world consists of quantum fields, with elementary events happening in spacetime [Rovelli] |
| 20459 | Electrons only exist when they interact, and their being is their combination of quantum leaps [Rovelli] |
| 20460 | Electrons are not waves, because their collisions are at a point, and not spread out [Rovelli] |
| 20466 | Quantum Theory describes events and possible interactions - not how things are [Rovelli] |
| 20465 | Nature has three aspects: granularity, indeterminacy, and relations [Rovelli] |
| 20458 | The world is just particles plus fields; space is the gravitational field [Rovelli] |
| 20470 | Only heat distinguishes past from future [Rovelli] |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |