Combining Texts

All the ideas for 'fragments/reports', 'Remarks on axiomatised set theory' and 'Letters to Voetius'

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6 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

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)

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

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)

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

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.

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism

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)

21. Aesthetics / C. Artistic Issues / 7. Art and Morality

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?

28. God / C. Attitudes to God / 5. Atheism

16712	Atheism is an atrocious and intolerable crime in any country [Descartes]
	Full Idea: Atheism is the most atrocious crime, and should be tolerated in no republic, however free.
	From: René Descartes (Letters to Voetius [1640], VIIIB:174), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2