6 ideas
17879 | Axiomatising set theory makes it all relative [Skolem] |
Full Idea: Axiomatising set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatisation. | |
From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.296) |
17878 | If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem] |
Full Idea: Löwenheim's theorem reads as follows: If a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain. | |
From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.293) |
17880 | Integers and induction are clear as foundations, but set-theory axioms certainly aren't [Skolem] |
Full Idea: The initial foundations should be immediately clear, natural and not open to question. This is satisfied by the notion of integer and by inductive inference, by it is not satisfied by the axioms of Zermelo, or anything else of that kind. | |
From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.299) | |
A reaction: This is a plea (endorsed by Almog) that the integers themselves should be taken as primitive and foundational. I would say that the idea of successor is more primitive than the integers. |
17881 | Mathematician want performable operations, not propositions about objects [Skolem] |
Full Idea: Most mathematicians want mathematics to deal, ultimately, with performable computing operations, and not to consist of formal propositions about objects called this or that. | |
From: Thoralf Skolem (Remarks on axiomatised set theory [1922], p.300) |
468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
22824 | Magna Carta forbids prison without trial, and insists on neutral and correct process [-, by Charvet] |
Full Idea: The Magna Carta forbids the King to imprison indefinitely without trial, and also binds the King to follow due process in his courts and not allow the justice provided to be for sale. | |
From: report of - (Magna Carta [1215]) by John Charvet - Liberalism: the basics 02 | |
A reaction: Very exasperating for a medieval monarch. In current times British law is exceedingly slow (so long imprisonment before trial), and the necessary effective advocates cost vastly too much for all but a tiny minority. So it's going badly. |