71 ideas
| 5728 | The concept of truth was originated by the senses [Lucretius] |
| 17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
| 17765 | Propositional language can only relate statements as the same or as different [Walicki] |
| 17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
| 17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
| 17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
| 3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
| 15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
| 3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
| 17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
| 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
| 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
| 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
| 17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
| 13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
| 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
| 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
| 17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
| 22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
| 17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
| 12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
| 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
| 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
| 17754 | Inductive proof depends on the choice of the ordering [Walicki] |
| 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
| 18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
| 15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
| 13672 | All the axioms for mathematics presuppose set theory [Neumann] |
| 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
| 5702 | The senses are much the best way to distinguish true from false [Lucretius] |
| 5729 | If the senses are deceptive, reason, which rests on them, is even worse [Lucretius] |
| 5697 | The only possible standard for settling doubts is the foundation of the senses [Lucretius] |
| 5727 | Most supposed delusions of the senses are really misinterpretations by the mind [Lucretius] |
| 5714 | Even simple facts are hard to believe at first hearing [Lucretius] |
| 5718 | The mind is in the middle of the breast, because there we experience fear and joy [Lucretius] |
| 5717 | The mind is a part of a man, just like a hand or an eye [Lucretius] |
| 21387 | The separate elements and capacities of a mind cannot be distinguished [Lucretius] |
| 5709 | The actions of the mind are not determinate and passive, because atoms can swerve [Lucretius] |
| 5695 | Only bodies can touch one another [Lucretius] |
| 5711 | The earth is and always has been an insentient being [Lucretius] |
| 5712 | Particles may have sensation, but eggs turning into chicks suggests otherwise [Lucretius] |
| 5719 | The mind moves limbs, wakes the body up, changes facial expressions, which involve touch [Lucretius] |
| 5724 | Lions, foxes and deer have distinct characters because their minds share in their bodies [Lucretius] |
| 5713 | You needn't be made of laughing particles to laugh, so why not sensation from senseless seeds? [Lucretius] |
| 6611 | One man's meat is another man's poison [Lucretius] |
| 5730 | Our bodies weren't created to be used; on the contrary, their creation makes a use possible [Lucretius] |
| 5726 | The dead are no different from those who were never born [Lucretius] |
| 5705 | Nature only wants two things: freedom from pain, and pleasure [Lucretius] |
| 5716 | Nature runs the universe by herself without the aid of gods [Lucretius] |
| 5704 | There can be no centre in infinity [Lucretius] |
| 5703 | The universe must be limitless, since there could be nothing outside to limit it [Lucretius] |
| 5693 | Everything is created and fed by nature from atoms, and they return to atoms in death [Lucretius] |
| 5701 | If an object is infinitely subdivisible, it will be the same as the whole universe [Lucretius] |
| 5708 | In downward motion, atoms occasionally swerve slightly for no reason [Lucretius] |
| 17004 | Nothing can break the binding laws of eternity [Lucretius] |
| 5696 | If there were no space there could be no movement, or even creation [Lucretius] |
| 5706 | Atoms move themselves [Lucretius] |
| 5700 | It is quicker to break things up than to assemble them [Lucretius] |
| 5698 | We can only sense time by means of movement, or its absence [Lucretius] |
| 5715 | This earth is very unlikely to be the only one created [Lucretius] |
| 5694 | Nothing can be created by divine power out of nothing [Lucretius] |
| 5699 | If matter wasn't everlasting, everything would have disappeared by now [Lucretius] |
| 5707 | The universe can't have been created by gods, because it is too imperfect [Lucretius] |
| 5710 | Gods are tranquil and aloof, and have no need of or interest in us [Lucretius] |
| 5731 | Why does Jupiter never hurl lightning from a blue sky? [Lucretius] |
| 5722 | For a separated spirit to remain sentient it would need sense organs attached to it [Lucretius] |
| 5725 | An immortal mind couldn't work harmoniously with a mortal body [Lucretius] |
| 5720 | Spirit is mortal [Lucretius] |
| 5721 | The mind is very small smooth particles, which evaporate at death [Lucretius] |
| 5723 | If spirit is immortal and enters us at birth, why don't we remember a previous existence? [Lucretius] |