38 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] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 22858 | There is collective action, where a trend is manifest, but is not attributable to individuals [Lukes] |
| 22854 | Power is a capacity, which may never need to be exercised [Lukes] |
| 22850 | Hidden powers are the most effective [Lukes] |
| 22852 | The pluralist view says that power is restrained by group rivalry [Lukes] |
| 22855 | One-dimensionsal power is behaviour in observable conflicts of interests [Lukes] |
| 22856 | Political organisation brings some conflicts to the fore, and suppresses others [Lukes] |
| 22857 | The two-dimensional view of power recognises the importance of controlling the agenda [Lukes] |
| 22859 | Power can be exercised to determine a person's desires [Lukes] |
| 22863 | Power is the capacity of a social class to realise its interests [Lukes] |
| 22860 | The evidence for the exertion of power need not involve a grievance of the powerless [Lukes] |
| 22861 | Power is affecting a person in a way contrary to their interests [Lukes] |
| 21133 | Supreme power is getting people to have thoughts and desires chosen by you [Lukes] |
| 22851 | In the 1950s they said ideology is finished, and expertise takes over [Lukes] |
| 22862 | Liberals take people as they are, and take their preferences to be their interests [Lukes] |
| 20575 | Socialist economics needs a very strong central power, virtually leading to slavery [Hayek, by Oksala] |
| 22853 | Anyone who thinks capitalism can improve their lives is endorsing capitalism [Lukes] |
| 22599 | Hayek was a liberal, but mainly concerned with market freedom [Hayek, by Dunt] |
| 22600 | Impeding the market is likely to lead to extensive state control [Hayek] |
| 20543 | 'Social justice' is a confused idea, and inequalities need no justification [Hayek, by Swift] |