42 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] |
| 22190 | If a theory is more informative it is less probable [Gorham] |
| 22189 | Why abandon a theory if you don't have a better one? [Gorham] |
| 22192 | Is Newton simpler with universal simultaneity, or Einstein simpler without absolute time? [Gorham] |
| 22194 | Structural Realism says mathematical structures persist after theory rejection [Gorham] |
| 22195 | Structural Realists must show the mathematics is both crucial and separate [Gorham] |
| 22197 | Theories aren't just for organising present experience if they concern the past or future [Gorham] |
| 22196 | For most scientists their concepts are not just useful, but are meant to be true and accurate [Gorham] |
| 22193 | Consilience makes the component sciences more likely [Gorham] |
| 22198 | Aristotelian physics has circular celestial motion and linear earthly motion [Gorham] |
| 21202 | The strong force has a considerably greater range than the weak force [Martin,BR] |
| 21211 | If an expected reaction does not occur, that implies a conservation law [Martin,BR] |
| 21209 | Electron emit and reabsorb photons, which create and reabsorb virtual electrons and positrons [Martin,BR] |
| 21201 | A 'field' is just a region to which points can be assigned in space and time [Martin,BR] |
| 21212 | The Higgs field, unlike others, has a nozero value in a state without particles [Martin,BR] |
| 21205 | Many physicists believe particles have further structure, if only we could see it [Martin,BR] |
| 21203 | Uncertainty allows very brief violations of energy conservation - even shorter with higher energies [Martin,BR] |
| 21207 | The Exclusion Principle says no two fermions occupy the same state, with the same numbers [Martin,BR] |
| 21204 | The standard model combines theories of strong interaction, and electromagnetic and weak interaction [Martin,BR] |
| 21208 | Eletrons don't literally 'spin', because they are point-like [Martin,BR] |
| 21210 | Virtual particles surround any charged particle [Martin,BR] |
| 21206 | The properties of a particle are determined by its quantum numbers and its mass [Martin,BR] |
| 21213 | String theory only has one free parameter (tension) - unlike the standard model with 19 [Martin,BR] |
| 21200 | An 'element' is what cannot be decomposed by chemistry [Martin,BR] |