8 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) |
| 21563 | The 'no classes' theory says the propositions just refer to the members [Russell] |
| Full Idea: The contention of the 'no classes' theory is that all significant propositions concerning classes can be regarded as propositions about all or some of their members. | |
| From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.200) | |
| A reaction: Apparently this theory has not found favour with later generations of theorists. I see it in terms of Russell trying to get ontology down to the minimum, in the spirit of Goodman and Quine. |
| 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) |
| 21565 | Richard's puzzle uses the notion of 'definition' - but that cannot be defined [Russell] |
| Full Idea: In Richard's puzzle, we use the notion of 'definition', and this, oddly enough, is not definable, and is indeed not a definite notion at all. | |
| From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.209) | |
| A reaction: The background for this claim is his type theory, which renders certain forms of circular reference meaningless. |
| 21564 | Vicious Circle: what involves ALL must not be one of those ALL [Russell] |
| Full Idea: The 'vicious-circle principle' says 'whatever involves an apparent variable must not be among the possible values of that variable', or (less exactly) 'whatever involves ALL must not be one of ALL which it involves. | |
| From: Bertrand Russell (On 'Insolubilia' and their solution [1906], p.204) | |
| A reaction: He offers this as a parallel to his 'no classes' principle. That referred to classes, but this refers to propositions, and specifically the Liar Paradox (which he calls the 'Epimenedes'). |
| 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? |