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) |
| 21963 | It is possible that an omnipotent God might make one and two fail to equal three [Descartes] |
| Full Idea: Since every basic truth depends on God's omnipotence, I would not dare to say that God cannot make it....that one and two should not be three. | |
| From: René Descartes (Letters to Antoine Arnauld [1645]), quoted by A.W. Moore - The Evolution of Modern Metaphysics 01.3 | |
| A reaction: An unusual view. Most people would say that if Descartes can doubt something that simple, he should also doubt his reasons for believing in God's existence. |
| 7566 | The Identity of Indiscernibles is really the same as the verification principle [Jolley] |
| Full Idea: Various writers have noted that the Identity of Indiscernibles is really tantamount to the verification principle. | |
| From: Nicholas Jolley (Leibniz [2005], Ch.3) | |
| A reaction: Both principles are false, because they are the classic confusion of epistemology and ontology. The fact that you cannot 'discern' a difference between two things doesn't mean that there is no difference. Things beyond verification can still be discussed. |