82 ideas
1887 | You cannot divide anything into many parts, because after the first division you are no longer dividing the original [Sext.Empiricus] |
1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
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] |
12196 | A valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood' [Sext.Empiricus] |
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] |
1902 | Since Socrates either died when he was alive (a contradiction) or died when he was dead (meaningless), he didn't die [Sext.Empiricus] |
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] |
17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
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] |
1889 | If an argument has an absurd conclusion, we should not assent to the absurdity, but avoid the absurd argument [Sext.Empiricus] |
1871 | Whether honey is essentially sweet may be doubted, as it is a matter of judgement rather than appearance [Sext.Empiricus] |
1883 | How can the intellect know if sensation is reliable if it doesn't directly see external objects? [Sext.Empiricus] |
1890 | We distinguish ambiguities by seeing what is useful [Sext.Empiricus] |
1870 | The basis of scepticism is the claim that every proposition has an equal opposing proposition [Sext.Empiricus] |
1882 | The necks of doves appear different in colour depending on the angle of viewing [Sext.Empiricus] |
1881 | The same oar seems bent in water and straight when out of it [Sext.Empiricus] |
1872 | The same tower appears round from a distance, but square close at hand [Sext.Empiricus] |
1873 | If we press the side of an eyeball, objects appear a different shape [Sext.Empiricus] |
1874 | How can we judge between our impressions and those of other animals, when we ourselves are involved? [Sext.Empiricus] |
1878 | Water that seems lukewarm can seem very hot on inflamed skin [Sext.Empiricus] |
1880 | Some actions seem shameful when sober but not when drunk [Sext.Empiricus] |
1877 | If we had no hearing or sight, we would assume no sound or sight exists, so there may be unsensed qualities [Sext.Empiricus] |
1879 | Sickness is perfectly natural to the sick, so their natural perceptions should carry some weight [Sext.Empiricus] |
1876 | If we enjoy different things, presumably we receive different impressions [Sext.Empiricus] |
1910 | With us it is shameful for men to wear earrings, but among Syrians it is considered noble [Sext.Empiricus] |
1911 | Even if all known nations agree on a practice, there may be unknown nations which disagree [Sext.Empiricus] |
1886 | If you don't view every particular, you may miss the one which disproves your universal induction [Sext.Empiricus] |
1884 | If we utter three steps of a logical argument, they never exist together [Sext.Empiricus] |
1894 | Some say that causes are physical, some say not [Sext.Empiricus] |
1897 | Knowing an effect results from a cause means knowing that the cause belongs with the effect, which is circular [Sext.Empiricus] |
1898 | Cause can't exist before effect, or exist at the same time, so it doesn't exist [Sext.Empiricus] |
1896 | If there were no causes then everything would have been randomly produced by everything [Sext.Empiricus] |
1895 | Causes are either equal to the effect, or they link equally with other causes, or they contribute slightly [Sext.Empiricus] |
1900 | If time and place are infinitely divided, it becomes impossible for movement ever to begin [Sext.Empiricus] |
1899 | Does the original self-mover push itself from behind, or pull itself from in front? [Sext.Empiricus] |
1901 | If all atoms, times and places are the same, everything should move with equal velocity [Sext.Empiricus] |
1903 | If motion and rest are abolished, so is time [Sext.Empiricus] |
1904 | Time must be unlimited, but past and present can't be non-existent, and can't be now, so time does not exist [Sext.Empiricus] |
1905 | How can time be divisible if we can't compare one length of time with another? [Sext.Empiricus] |
1891 | How can we agree on the concept of God, unless we agree on his substance or form or place? [Sext.Empiricus] |
1892 | The existence of God can't be self-evident or everyone would have agreed on it, so it needs demonstration [Sext.Empiricus] |
1893 | If God foresaw evil he would presumably prevent it, and if he only foresees some things, why those things? [Sext.Empiricus] |