30 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] |
| 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] |
| 2667 | A false object might give the same presentation as a true one [Arcesilaus, by Cicero] |
| 7621 | Special relativity, unlike general relativity, was operationalist in spirit [Putnam on Einstein] |
| 22955 | Einstein took causation to be the bedrock of physics [Einstein, by Coveney/Highfield] |
| 20638 | General relativity assumes laws of nature are the same in all frames of reference [Einstein, by Close] |
| 20636 | Newton is a special case of Einstein's general theory, with an infinite speed of light [Einstein, by Close] |
| 21230 | The theory is 'special' because it sticks to observers moving straight, at constant speeds [Einstein, by Farmelo] |
| 21231 | Assume the speed of light is constant for all observers, and the laws of physics are the same [Einstein, by Farmelo] |
| 20634 | General Relativity says there is no absolute force or acceleration [Einstein, by Close] |
| 20648 | Mass is a measure of energy content [Einstein] |
| 21232 | Space-time arises from the connection between measurements of space and of time [Einstein, by Farmelo] |
| 7626 | I do not believe in a personal God [Einstein] |