100 ideas
13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
13477 | The problems are the monuments of philosophy [Hart,WD] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
19767 | Reason leads to prudent selfishness, which overrules natural compassion [Rousseau] |
13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
19757 | No one would bother to reason, and try to know things, without a desire for enjoyment [Rousseau] |
13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
19760 | General ideas are purely intellectual; imagining them is immediately particular [Rousseau] |
19759 | Only words can introduce general ideas into the mind [Rousseau] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
19758 | Language may aid thinking, but powerful thought was needed to produce language [Rousseau] |
19773 | Without love, what use is beauty? [Rousseau] |
19769 | Rational morality is OK for brainy people, but ordinary life can't rely on that [Rousseau] |
19752 | If we should not mistreat humans, it is mainly because of sentience, not rationality [Rousseau] |
19768 | The better Golden Rule is 'do good for yourself without harming others' [Rousseau] |
19766 | The fact that we weep (e.g. in theatres) shows that we are naturally compassionate [Rousseau] |
19756 | Humans are less distinguished from other animals by understanding, than by being free agents [Rousseau] |
19755 | Most human ills are self-inflicted; the simple, solitary, regular natural life is good [Rousseau] |
19762 | Is language a pre-requisite for society, or might it emerge afterwards? [Rousseau] |
19763 | I doubt whether a savage person ever complains of life, or considers suicide [Rousseau] |
19778 | Leisure led to envy, inequality, vice and revenge, which we now see in savages [Rousseau] |
19779 | Primitive man was very gentle [Rousseau] |
19751 | Our two starting principles are concern for self-interest, and compassion for others [Rousseau] |
19765 | Savages avoid evil because they are calm, and never think of it (not because they know goodness) [Rousseau] |
19771 | Savage men quietly pursue desires, without the havoc of modern frenzied imagination [Rousseau] |
19774 | A savage can steal fruit or a home, but there is no means of achieving obedience [Rousseau] |
19772 | In a state of nature people are much more equal; it is society which increases inequalities [Rousseau] |
19789 | It is against nature for children to rule old men, fools to rule the wise, and the rich to hog resources [Rousseau] |
19787 | People accept the right to be commanded, because they themselves wish to command [Rousseau] |
19780 | We seem to have made individual progress since savagery, but actually the species has decayed [Rousseau] |
19747 | Revolutionaries usually confuse liberty with total freedom, and end up with heavier chains [Rousseau] |
19748 | Plebiscites are bad, because they exclude the leaders from crucial decisions [Rousseau] |
19749 | In a direct democracy, only the leaders should be able to propose new laws [Rousseau] |
19784 | Enslaved peoples often boast of their condition, calling it a state of 'peace' [Rousseau] |
19785 | If the child of a slave woman is born a slave, then a man is not born a man [Rousseau] |
19775 | People must be made dependent before they can be enslaved [Rousseau] |
19746 | Like rich food, liberty can ruin people who are too weak to cope with it [Rousseau] |
19786 | Three stages of the state produce inequalities of wealth, power, and enslavement [Rousseau] |
19788 | The pleasure of wealth and power is largely seeing others deprived of them [Rousseau] |
19777 | Persuading other people that some land was 'owned' was the beginning of society [Rousseau] |
19782 | What else could property arise from, but the labour people add to it? [Rousseau] |
19781 | Land cultivation led to a general right of ownership, administered justly [Rousseau] |
19754 | If we have a natural right to property, what exactly does 'belonging to' mean? [Rousseau] |
19750 | Writers just propose natural law as the likely useful agreements among people [Rousseau] |
19770 | Primitive people simply redressed the evil caused by violence, without thought of punishing [Rousseau] |
19783 | A state of war remains after a conquest, if the losers don't accept the winners [Rousseau] |
19753 | Both men and animals are sentient, which should give the latter the right not to be mistreated [Rousseau] |
19761 | Men started with too few particular names, but later had too few natural kind names [Rousseau] |
19776 | Small uninterrupted causes can have big effects [Rousseau] |