33 ideas
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| 16554 | Activities have place, rate, duration, entities, properties, modes, direction, polarity, energy and range [Machamer/Darden/Craver] |
| 16556 | Penicillin causes nothing; the cause is what penicillin does [Machamer/Darden/Craver] |
| 472 | No things would be clear to us as entity or relationships unless there existed Number and its essence [Philolaus] |
| 16562 | We understand something by presenting its low-level entities and activities [Machamer/Darden/Craver] |
| 16563 | The explanation is not the regularity, but the activity sustaining it [Machamer/Darden/Craver] |
| 16555 | Functions are not properties of objects, they are activities contributing to mechanisms [Machamer/Darden/Craver] |
| 16529 | Mechanisms are systems organised to produce regular change [Machamer/Darden/Craver] |
| 16530 | A mechanism explains a phenomenon by showing how it was produced [Machamer/Darden/Craver] |
| 16553 | Our account of mechanism combines both entities and activities [Machamer/Darden/Craver] |
| 16559 | Descriptions of explanatory mechanisms have a bottom level, where going further is irrelevant [Machamer/Darden/Craver] |
| 16528 | Mechanisms are not just push-pull systems [Machamer/Darden/Craver] |
| 16564 | There are four types of bottom-level activities which will explain phenomena [Machamer/Darden/Craver] |
| 16561 | We can abstract by taking an exemplary case and ignoring the detail [Machamer/Darden/Craver] |
| 22511 | Some reasonings are stronger than we are [Philolaus] |
| 473 | There is no falsehood in harmony and number, only in irrational things [Philolaus] |
| 1519 | Harmony must pre-exist the cosmos, to bring the dissimilar sources together [Philolaus] |
| 1518 | Everything must involve numbers, or it couldn't be thought about or known [Philolaus] |
| 469 | Existing things, and hence the Cosmos, are a mixture of the Limited and the Unlimited [Philolaus] |
| 476 | Self-created numbers make the universe stable [Philolaus] |
| 16558 | Laws of nature have very little application in biology [Machamer/Darden/Craver] |
| 1787 | Philolaus was the first person to say the earth moves in a circle [Philolaus, by Diog. Laertius] |