99 ideas
| 20910 | Everything happens necessarily, and for a reason [Democritus] |
| 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] |
| 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] |
| 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] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [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] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [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] |
| 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] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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] |
| 16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
| 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] |
| 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] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 20901 | True Being only occurs when it is completely full, with atoms and no void [Democritus, by Aristotle] |
| 20902 | Being does not exist more than non-being [Democritus, by Aristotle] |
| 20904 | The non-existent exists as much as the existent, because it has causal powers [Democritus] |
| 20903 | The only distinctions are Configuration (shape), Disposition (order) and Turning (position) [Democritus, by Aristotle] |
| 20893 | Nothing comes from non-existence, or passes into it [Democritus, by Diog. Laertius] |
| 20896 | It is not possible to know what sort each thing is [Democritus] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 3357 | Democritus denies reality to large objects, because atomic entities can't combine to produce new ones [Benardete,JA on Democritus] |
| 598 | Democritus said that substances could never be mixed, so atoms are the substances [Democritus, by Aristotle] |
| 1532 | Sensible qualities can't be real if they appear different to different creatures [Democritus, by Theophrastus] |
| 20894 | Man is separated from reality [Democritus] |
| 517 | All evidence comes from senses, so they are indispensable to the mind [Democritus] |
| 20897 | Obscure knowledge belongs to the five senses, and genuine knowledge is the other type [Democritus] |
| 577 | Democritus says there is either no truth, or it is concealed from us [Democritus, by Aristotle] |
| 20895 | We actually know nothing, and opinions are mere flux [Democritus] |
| 1528 | We in fact know nothing, but we each restructure our reality with beliefs [Democritus] |
| 492 | It is obviously impossible to understand the reality of each thing [Democritus] |
| 515 | We know nothing in reality; for truth lies in an abyss [Democritus] |
| 20892 | Democritus was devoted to discovering causal explanations [Democritus, by Eusebius] |
| 5882 | Democritus says soul consists of smooth round bodies brought together in accidental collision [Democritus, by Cicero] |
| 6034 | Atomists say soul has a rational part in the chest, and a diffused non-rational part [Democritus, by Aetius] |
| 20912 | The soul is the same as the mind [Democritus, by Aristotle] |
| 20916 | Animals have a share of reason [Democritus, by Porphyry] |
| 20914 | The directive centre is located in the whole head [Democritus, by Ps-Plutarch] |
| 6033 | Democritus said everything happens of necessity, by natural motion of atoms [Democritus, by Cicero] |
| 5088 | Some say there is a determinate cause for every apparently spontaneous event [Democritus, by Aristotle] |
| 21670 | Democritus said atoms only move by their natural motions, which are therefore necessary [Democritus, by Cicero] |
| 24041 | Democritus says spherical atoms are fire, and constitute the soul [psuche] [Democritus, by Aristotle] |
| 20913 | Democritus says the soul is the body, and thinking is thus the mixture of the body [Democritus, by Theophrastus] |
| 2604 | We must have expressive power BEFORE we learn language [Fodor] |
| 1540 | Pleasure and pain guide our choices of good and bad [Democritus] |
| 495 | Wisdom creates a healthy passion-free soul [Democritus] |
| 1537 | Happiness is identifying and separating the pleasures [Democritus, by Stobaeus] |
| 20917 | Contentment comes from moderation and proportion in life [Democritus, by Stobaeus] |
| 13551 | Democritus says wealth is a burden to the virtuous mind [Democritus, by Seneca] |
| 20899 | Atoms cling together, until a stronger necessity disperses them [Democritus, by Aristotle] |
| 20898 | Atoms are irregular, hooked, concave, convex, and many other shapes [Democritus, by Aristotle] |
| 17542 | 'Full' and 'Void' secularised Parmenides's Being and Not-being [Democritus, by Heisenberg] |
| 1525 | Atomists say there are only three differences - in shape, arrangement and position [Democritus, by Aristotle] |
| 493 | Experiences are merely convention; only atoms and the void are real [Democritus] |
| 5947 | If only atoms are real and the rest is convention, we wouldn't bother to avoid pain [Democritus, by Diogenes of Oen.] |
| 13219 | When atoms touch, why don't they coalesce, like water drops? [Aristotle on Democritus] |
| 1533 | Because appearance is infinitely varied, atomists assume infinitely many shapes of atom [Democritus, by Aristotle] |
| 20908 | There could be an atom the size of the world [Democritus, by Ps-Plutarch] |
| 1527 | There must be atoms, to avoid the absurdity of infinite division down to nothing [Democritus, by Aristotle] |
| 20909 | The basic atoms are without qualities - which only arise from encounters between atoms [Democritus, by Galen] |
| 1536 | If a cone is horizontally sliced the surfaces can't be equal, so it goes up in steps [Democritus] |
| 23314 | Greeks explained regularity by intellectual design, not by laws [Democritus, by Frede,M] |
| 20905 | Growth and movement would not exist if there were no void to receive them [Democritus] |
| 24059 | Democritus is wrong: in a void we wouldn't see a distant ant in exact detail [Aristotle on Democritus] |
| 5101 | Movement is impossible in a void, because nothing can decide the direction of movement [Aristotle on Democritus] |
| 20911 | There are unlimited worlds of varying sizes, some without life or water [Democritus, by Hippolytus] |
| 1535 | Democritus said people imagined gods as the source of what awed or frightened them [Democritus, by Sext.Empiricus] |
| 20915 | The soul is destroyed with the body [Democritus, by Ps-Plutarch] |