display all the ideas for this combination of texts
4 ideas
| 562 | Axioms are the underlying principles of everything, and who but the philosopher can assess their truth? [Aristotle] |
| Full Idea: Axioms are more general, and the principles of all things. If this does not belong to the philosopher, who else will have the job of considering truth and falsity in their case? | |
| From: Aristotle (Metaphysics [c.324 BCE], 0997a09) |
| 573 | The axioms of mathematics are part of philosophy [Aristotle] |
| Full Idea: A single science, that of the philosopher, also covers the axioms of mathematics. | |
| From: Aristotle (Metaphysics [c.324 BCE], 1005a15) |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |