41 ideas
| 2922 | All intelligent Romans were Epicureans [Nietzsche] |
| 23520 | Truth has had to be fought for, and normal life must be sacrificed to achieve it [Nietzsche] |
| 2914 | One must never ask whether truth is useful [Nietzsche] |
| 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] |
| 2921 | Philosophy grasps the limits of human reason, and values are beyond it [Nietzsche] |
| 20138 | Christianity is at war with the higher type of man, and excommunicates his basic instincts [Nietzsche] |
| 20375 | Virtues must be highly personal; if not, it is merely respect for a concept [Nietzsche] |
| 2915 | Each person should devise his own virtues and categorical imperative [Nietzsche] |
| 20086 | Nowadays sovereignty (once the basis of a state) has become relative [Reybrouck] |
| 20090 | Today it seems almost impossible to learn the will of the people [Reybrouck] |
| 20087 | There are no united monolothic 'peoples', and no 'national gut feelings' [Reybrouck] |
| 20089 | Technocrats may be efficient, but they lose legitimacy as soon as they do unpopular things [Reybrouck] |
| 20088 | Technocrats are expert managers, who replace politicians, and can be long-term and unpopular [Reybrouck] |
| 20085 | Democracy is the best compromise between legitimacy and efficiency [Reybrouck] |
| 20095 | A referendum result arises largely from ignorance [Reybrouck] |
| 20094 | You don't really govern people if you don't involve them [Reybrouck] |
| 20093 | In the 18th century democratic lots lost out to elections, that gave us a non-hereditary aristocracy [Reybrouck] |
| 20091 | Representative elections were developed in order to avoid democracy [Reybrouck] |
| 2920 | A God who cures us of a head cold at the right moment is a total absurdity [Nietzsche] |
| 2917 | Christianity is a revolt of things crawling on the ground against elevated things [Nietzsche] |
| 2918 | The story in Genesis is the story of God's fear of science [Nietzsche] |
| 2919 | 'Faith' means not wanting to know what is true [Nietzsche] |
| 2916 | The great lie of immortality destroys rationality and natural instinct [Nietzsche] |