72 ideas
| 5278 | Philosophy is no more than abstractions concerning observations of human historical development [Marx/Engels] |
| 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] |
| 5287 | Philosophical problems are resolved into empirical facts [Marx/Engels] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 23872 | 'Society determines consciousness' is contradictory; society only exists in minds [Weil on Marx/Engels] |
| 5277 | Life is not determined by consciousness, but consciousness by life [Marx/Engels] |
| 5280 | Language co-exists with consciousness, and makes it social [Marx/Engels] |
| 5276 | The nature of an individual coincides with what they produce and how they produce it [Marx/Engels] |
| 5281 | Consciousness is a social product [Marx/Engels] |
| 5289 | When aristocracy or the bourgeoisie dominate, certain values dominate with them [Marx/Engels] |
| 5274 | Young Hegelians proposed changing our present consciousness for liberating critical consciousness [Marx/Engels] |
| 21992 | Producing their own subsistence distinguishes men from animals [Marx/Engels] |
| 5275 | Men distinguish themselves from animals when they begin to produce their means of subsistence [Marx/Engels] |
| 5292 | Individuals are mutually hostile unless they group together in competition with other groups [Marx/Engels] |
| 20582 | World government needs a shared global identity [Oksala] |
| 20568 | The principles Rawls arrives at do not just conform to benevolence, but also result from choices [Oksala] |
| 20578 | Anarchists prefer local and communal government [Oksala] |
| 20574 | Utilitarianism neglects responsibility, duties and rights [Oksala] |
| 5293 | Only in community are people able to cultivate their gifts, and therefore be free [Marx/Engels] |
| 5273 | Young Hegelians think consciousness is chains for men, where old Hegelians think it the bond of society [Marx/Engels] |
| 5283 | In communist society we are not trapped in one activity, but can act freely [Marx/Engels] |
| 5282 | If the common interest imposes on the individual, his actions become alienated and enslaving [Marx/Engels] |
| 5288 | The class controlling material production also controls mental production [Marx/Engels] |
| 5290 | The revolutionary class is opposed to 'class', and represents all of society [Marx/Engels] |
| 5294 | To assert themselves as individuals, the proletarians must overthrow the State [Marx/Engels] |
| 5285 | Slavery cannot be abolished without the steam-engine [Marx/Engels] |
| 5284 | Communism abolishes private property and dissolves the powerful world market [Marx/Engels] |
| 5291 | The law says private property is the result of the general will [Marx/Engels] |
| 5279 | Human history must always be studied in relation to industry and exchange [Marx/Engels] |
| 5286 | Most historians are trapped in the illusions of their own epoch [Marx/Engels] |