49 ideas
| 192 | Only one thing can be contrary to something [Plato] |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| 190 | If asked whether justice itself is just or unjust, you would have to say that it is just [Plato] |
| 20185 | The most important things in life are wisdom and knowledge [Plato] |
| 20184 | The only real evil is loss of knowledge [Plato] |
| 191 | Everything resembles everything else up to a point [Plato] |
| 203 | Courage is knowing what should or shouldn't be feared [Plato] |
| 202 | No one willingly and knowingly embraces evil [Plato] |
| 193 | Some things are good even though they are not beneficial to men [Plato] |
| 197 | Some pleasures are not good, and some pains are not evil [Plato] |
| 200 | People tend only to disapprove of pleasure if it leads to pain, or prevents future pleasure [Plato] |
| 188 | Socrates did not believe that virtue could be taught [Plato] |
| 204 | Socrates is contradicting himself in claiming virtue can't be taught, but that it is knowledge [Plato] |
| 189 | If we punish wrong-doers, it shows that we believe virtue can be taught [Plato] |
| 7337 | In Mosaic legal theory, crimes are sins and sins are crimes [Johnson,P] |
| 7339 | Because human life is what is sacred, Mosaic law has no death penalty for property violations [Johnson,P] |
| 7353 | The Pharisees undermined slavery, by giving slaves responsibility and status in law courts [Johnson,P] |
| 7340 | Mosaic law was the first to embody the rule of law, and equality before the law [Johnson,P] |
| 7338 | Man's life is sacred, because it is made in God's image [Johnson,P] |
| 7348 | The Jews sharply distinguish human and divine, but the Greeks pull them closer together [Johnson,P] |
| 7336 | A key moment is the idea of a single moral God, who imposes his morality on humanity [Johnson,P] |
| 7341 | Sampson illustrates the idea that religious heroes often begin as outlaws and semi-criminals [Johnson,P] |
| 7342 | Isaiah moved Israelite religion away from the local, onto a more universal plane [Johnson,P] |
| 7355 | The Torah pre-existed creation, and was its blueprint [Johnson,P] |
| 7344 | Judaism involves circumcision, Sabbath, Passover, Pentecost, Tabernacles, New Year, and Atonement [Johnson,P] |
| 7345 | In exile the Jews became a nomocracy [Johnson,P] |
| 7347 | Zoroastrians believed in one eternal beneficent being, Creator through the holy spirit [Johnson,P] |
| 7349 | Immortality based on judgement of merit was developed by the Egyptians (not the Jews) [Johnson,P] |
| 7354 | The main doctrine of the Pharisees was belief in resurrection and the afterlife [Johnson,P] |
| 7356 | Pious Jews saw heaven as a vast library [Johnson,P] |