77 ideas
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 15845 | It seems absurd that seeing a person's limbs, the one is many, and yet the many are one [Plato] |
| 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] |
| 9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
| 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] |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| 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] |
| 15857 | Any mixture which lacks measure and proportion doesn't even count as a mixture at all [Plato] |
| 14503 | If a mixture does not contain measure and proportion, it is corrupted and destroyed [Plato] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 4447 | If the good is one, is it unchanged when it is in particulars, and is it then separated from itself? [Plato] |
| 15856 | A thing can become one or many, depending on how we talk about it [Plato] |
| 374 | If one object is divided into its parts, someone can then say that one are many and many is one [Plato] |
| 389 | How can you be certain about aspects of the world if they aren't constant? [Plato] |
| 390 | If goodness involves moderation and proportion, then it seems to be found in beauty [Plato] |
| 392 | Neither intellect nor pleasure are the good, because they are not perfect and self-sufficient [Plato] |
| 391 | The good involves beauty, proportion and truth [Plato] |
| 393 | Good first, then beauty, then reason, then knowledge, then pleasure [Plato, by PG] |
| 385 | Some of the pleasures and pains we feel are false [Plato] |
| 387 | A small pure pleasure is much finer than a large one contaminated with pain [Plato] |
| 373 | Pleasure is certainly very pleasant, but it doesn't follow that all pleasures are good [Plato] |
| 379 | The good must be sufficient and perfect, and neither intellect nor pleasure are that [Plato] |
| 371 | Reason, memory, truth and wisdom are far better than pleasure, for those who can attain them [Plato] |
| 376 | Would you prefer a life of pleasure without reason, or one of reason without pleasure? [Plato] |
| 382 | It is unlikely that the gods feel either pleasure or pain [Plato] |
| 381 | We feel pleasure when we approach our natural state of harmony [Plato] |
| 386 | Intense pleasure and pain are not felt in a good body, but in a worthless one [Plato] |
| 388 | Hedonists must say that someone in pain is bad, even if they are virtuous [Plato] |
| 377 | If you lived a life of maximum pleasure, would you still be lacking anything? [Plato] |
| 378 | A life of pure pleasure with no intellect is the life of a jellyfish [Plato] |
| 19855 | The purpose of society is to protect the rights of liberty, property, security and resistance [Mirabeau/committee] |
| 19856 | The law expresses the general will, and all citizens can participate [Mirabeau/committee] |
| 19861 | There is only a constitution if rights are assured, and separation of powers defined [Mirabeau/committee] |
| 19858 | No one should be molested for their opinions, if they do not disturb the established order [Mirabeau/committee] |
| 19859 | Free speech is very precious, and everyone may speak and write freely (but take responsibility for it) [Mirabeau/committee] |
| 19857 | All citizens are eligible for roles in the state, purely on the basis of merit [Mirabeau/committee] |
| 19862 | Property is a sacred right, breached only when essential, and with fair compensation [Mirabeau/committee] |
| 19860 | Everyone must contribute to the state's power and administration, in just proportion [Mirabeau/committee] |