68 ideas
18290 | But what is the reasoning of the body, that it requires the wisdom you seek? [Nietzsche] |
18303 | Reject wisdom that lacks laughter [Nietzsche] |
18305 | To love truth, you must know how to lie [Nietzsche] |
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] |
10066 | Putnam coined the term 'if-thenism' [Putnam, by Musgrave] |
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] |
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] |
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] |
10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
10852 | Robinson Arithmetic (Q) is not negation complete [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] |
20757 | The powerful self behind your thoughts and feelings is your body [Nietzsche] |
18289 | Forget the word 'I'; 'I' is performed by the intelligence of your body [Nietzsche] |
18299 | The will is constantly frustrated by the past [Nietzsche] |
18297 | We created meanings, to maintain ourselves [Nietzsche] |
18293 | The noble man wants new virtues; the good man preserves what is old [Nietzsche] |
18301 | We only really love children and work [Nietzsche] |
18307 | I want my work, not happiness! [Nietzsche] |
18291 | Virtues can destroy one another, through jealousy [Nietzsche] |
18287 | People now find both wealth and poverty too much of a burden [Nietzsche] |
18295 | If you want friends, you must be a fighter [Nietzsche] |
18286 | The greatest experience possible is contempt for your own happiness, reason and virtue [Nietzsche] |
18296 | An enduring people needs its own individual values [Nietzsche] |
18294 | The state coldly claims that it is the people, but that is a lie [Nietzsche] |
18304 | Saints want to live as they desire, or not to live at all [Nietzsche] |
18300 | Whenever we have seen suffering, we have wanted the revenge of punishment [Nietzsche] |
18302 | Man and woman are deeply strange to one another! [Nietzsche] |
18292 | I can only believe in a God who can dance [Nietzsche] |
18298 | Not being a god is insupportable, so there are no gods! [Nietzsche] |
18288 | Heaven was invented by the sick and the dying [Nietzsche] |
18306 | We don't want heaven; now that we are men, we want the kingdom of earth [Nietzsche] |