78 ideas
| 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] |
| 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] |
| 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] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 4266 | Having beliefs involves recognition, expectation and surprise [Scruton] |
| 4265 | If an animal has beliefs, that implies not only that it can make mistakes, but that it can learn from them [Scruton] |
| 4264 | Perception (which involves an assessment) is a higher state than sensation [Scruton] |
| 4271 | There is consciousness whenever behaviour must be explained in terms of mental activity [Scruton] |
| 4272 | Our concept of a person is derived from Roman law [Scruton] |
| 4267 | Conditioning may change behaviour without changing the mind [Scruton] |
| 4269 | An emotion is a motive which is also a feeling [Scruton] |
| 4270 | Do we use reason to distinguish people from animals, or use that difference to define reason? [Scruton] |
| 19045 | Translation is too flimsy a notion to support theories of cultural incommensurability [Quine] |
| 4284 | All moral life depends ultimately on piety, which is our recognition of our own dependence [Scruton] |
| 4273 | Kant's Moral Law is the rules rational beings would accept when trying to live by agreement [Scruton] |
| 4274 | The modern virtues are courage, prudence, wisdom, temperance, justice, charity and loyalty [Scruton] |
| 4286 | Only just people will drop their own self-interests when faced with an impartial verdict [Scruton] |
| 4283 | Sympathy can undermine the moral order just as much as crime does [Scruton] |
| 4290 | That which can only be done by a callous person, ought not to be done [Scruton] |
| 4285 | As soon as we drop self-interest and judge impartially, we find ourselves agreeing about conflicts [Scruton] |
| 4287 | Utilitarianism merely guides us (by means of sympathy) when the moral law is silent [Scruton] |
| 4282 | Morality is not a sort of calculation, it is what sets the limits to when calculation is appropriate [Scruton] |
| 4281 | Utilitarianism says we can't blame Stalin yet, but such a theory is a sick joke [Scruton] |
| 4280 | Utilitarianism is wrong precisely because it can't distinguish animals from people [Scruton] |
| 4296 | Brutal animal sports are banned because they harm the personality of the watcher [Scruton] |
| 4288 | Many breeds of animals have needs which our own ancestors planted in them [Scruton] |
| 4293 | Introducing a natural means of controlling animal population may not be very compassionate [Scruton] |
| 4295 | We favour our own animals over foreign ones because we see them as fellow citizens [Scruton] |
| 4268 | Animals command our sympathy and moral concern initially because of their intentionality [Scruton] |
| 4291 | Letting your dog kill wild rats, and keeping rats for your dog to kill, are very different [Scruton] |
| 4263 | Many of the stranger forms of life (e.g. worms) interest us only as a species, not as individuals [Scruton] |
| 4276 | An animal has individuality if it is nameable, and advanced animals can respond to their name [Scruton] |
| 4277 | I may avoid stepping on a spider or flower, but fellow-feeling makes me protect a rabbit [Scruton] |
| 4278 | Lucky animals are eaten by large predators, the less lucky starve, and worst is death by small predators [Scruton] |
| 4279 | We can easily remove the risk of suffering from an animal's life, but we shouldn't do it [Scruton] |
| 4289 | Sheep and cattle live comfortable lives, and die an enviably easy death [Scruton] |
| 4292 | Concern for one animal may harm the species, if the individual is part of a bigger problem [Scruton] |
| 4294 | Animals are outside the community of rights, but we still have duties towards them [Scruton] |