21 ideas
10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
21243 | An existing thing is even greater if its non-existence is inconceivable [Anselm] |
21244 | Conceiving a greater being than God leads to absurdity [Anselm] |
21241 | Even the fool can hold 'a being than which none greater exists' in his understanding [Anselm] |
21242 | If that than which a greater cannot be thought actually exists, that is greater than the mere idea [Anselm] |
1421 | A perfection must be independent and unlimited, and the necessary existence of Anselm's second proof gives this [Malcolm on Anselm] |
21245 | The word 'God' can be denied, but understanding shows God must exist [Anselm] |
21246 | Guanilo says a supremely fertile island must exist, just because we can conceive it [Anselm] |
21247 | Nonexistence is impossible for the greatest thinkable thing, which has no beginning or end [Anselm] |
1420 | Anselm's first proof fails because existence isn't a real predicate, so it can't be a perfection [Malcolm on Anselm] |