75 ideas
21422 | Moral self-knowledge is the beginning of all human wisdom [Kant] |
21408 | For any subject, its system of non-experiential concepts needs a metaphysics [Kant] |
21416 | Philosophers should not offer multiple proofs - suggesting the weakness of each of them [Kant] |
10073 | There cannot be a set theory which is complete [Smith,P] |
10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
10605 | Two functions are the same if they have the same extension [Smith,P] |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
10613 | No nice theory can define truth for its own language [Smith,P] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
21410 | That a concept is not self-contradictory does not make what it represents possible [Kant] |
21421 | Within nature man is unimportant, but as moral person he is above any price [Kant] |
21415 | Duty is impossible without prior moral feeling, conscience, love and self-respect [Kant] |
21409 | Moral principles do not involve feelings [Kant] |
21431 | The love of man is required in order to present the world as a beautiful and perfect moral whole [Kant] |
21437 | All morality directs the will to love of others' ends, and respect for others' rights [Kant] |
21429 | The duty of love is to makes the ends of others one's own [Kant] |
21411 | A duty of virtue is a duty which is also an end [Kant] |
21413 | Virtue is strong maxims for duty [Kant] |
21414 | The supreme principle of virtue is to find universal laws for ends [Kant] |
21436 | We are obliged to show the social virtues, but at least they make a virtuous disposition fashionable [Kant] |
21419 | If virtue becomes a habit, that is a loss of the freedom needed for adopting maxims [Kant] |
21417 | How do we distinguish a mean? The extremes can involve quite different maxims [Kant] |
21420 | If virtue is the mean between vices, then virtue is just the vanishing of vice [Kant] |
21418 | There is one principle of virtues; the virtues are distinguished by their objects [Kant] |
21425 | We can love without respect, and show respect without love [Kant] |
21427 | Respect is limiting our self-esteem by attending to the human dignity of other persons [Kant] |
21430 | Disrespect is using a person as a mere means to my own ends [Kant] |
21428 | Respect is purely negative (of not exalting oneself over others), and is thus a duty of Right [Kant] |
21426 | Love urges us to get closer to people, but respect to keep our distance [Kant] |
21434 | We must respect the humanity even in a vicious criminal [Kant] |
21412 | Humans are distinguished from animals by their capacity to set themselves any sort of end [Kant] |
21435 | Man is both social, and unsociable [Kant] |
21433 | Violation of rights deserves punishment, which is vengeance, rather than restitution [Kant] |
21423 | Men can only have duties to those who qualify as persons [Kant] |
21424 | Cruelty to animals is bad because it dulls our empathy for pain in humans [Kant] |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |