16 ideas
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
10282 | Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W] |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
10283 | A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W] |
10284 | There are three different standard presentations of semantics [Hodges,W] |
10285 | I |= φ means that the formula φ is true in the interpretation I [Hodges,W] |
10288 | Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W] |
10289 | Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W] |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
10287 | If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W] |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
10286 | A 'set' is a mathematically well-behaved class [Hodges,W] |
8732 | It is spooky the way mathematics anticipates physics [Weinberg] |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |