52 ideas
| 21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| 17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| 10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
| 17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
| 10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
| 17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
| 10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
| 9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
| 3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
| 11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
| 10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
| 10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
| 10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
| 10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| 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] |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| 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] |
| 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] |
| 4288 | Many breeds of animals have needs which our own ancestors planted in them [Scruton] |
| 4295 | We favour our own animals over foreign ones because we see them as fellow citizens [Scruton] |
| 4296 | Brutal animal sports are banned because they harm the personality of the watcher [Scruton] |
| 4291 | Letting your dog kill wild rats, and keeping rats for your dog to kill, are very different [Scruton] |
| 4293 | Introducing a natural means of controlling animal population may not be very compassionate [Scruton] |
| 4268 | Animals command our sympathy and moral concern initially because of their intentionality [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] |