32 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] |
| 17486 | Supervenience is simply modally robust property co-variance [Hendry] |
| 17481 | Nuclear charge (plus laws) explains electron structure and spectrum, but not vice versa [Hendry] |
| 17478 | Maybe two kinds are the same if there is no change of entropy on isothermal mixing [Hendry] |
| 17484 | Maybe the nature of water is macroscopic, and not in the microstructure [Hendry] |
| 17479 | The nature of an element must survive chemical change, so it is the nucleus, not the electrons [Hendry] |
| 17485 | Maybe water is the smallest part of it that still counts as water (which is H2O molecules) [Hendry] |
| 17482 | Compounds can differ with the same collection of atoms, so structure matters too [Hendry] |
| 17483 | Water continuously changes, with new groupings of molecules [Hendry] |
| 17476 | Elements survive chemical change, and are tracked to explain direction and properties [Hendry] |
| 17477 | Defining elements by atomic number allowed atoms of an element to have different masses [Hendry] |
| 17480 | Generally it is nuclear charge (not nuclear mass) which determines behaviour [Hendry] |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| 6610 | I believe because it is absurd [Tertullian] |