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All the ideas for 'poems', 'works' and 'Introduction to Zermelo's 1930 paper'

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6 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17833 The first-order ZF axiomatisation is highly non-categorical [Hallett,M]
     Full Idea: The first-order Sermelo-Fraenkel axiomatisation is highly non-categorical.
     From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1213)
17834 Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M]
     Full Idea: The non-categoricity of the axioms which Zermelo demonstrates reveals an incompleteness of a sort, ....for this seems to show that there will always be a set (indeed, an unending sequence) that the basic axioms are incapable of revealing to be sets.
     From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1215)
     A reaction: Hallett says the incompleteness concerning Zermelo was the (transfinitely) indefinite iterability of the power set operation (which is what drives the 'iterative conception' of sets).
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
17837 Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M]
     Full Idea: Unlike earlier writers (such as Fraenkel), Zermelo clearly allows that there might be ur-elements (that is, objects other than the empty set, which have no members). Indeed he sees in this the possibility of widespread application of set-theory.
     From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17836 The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]
     Full Idea: In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself.
     From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217)
21. Aesthetics / B. Nature of Art / 4. Art as Expression
8120 Objects can be beautiful which express nothing at all, such as the rainbow [Herbart, by Tolstoy]
     Full Idea: Objects are often beautiful which express nothing at all, as, for instance, the rainbow, which is beautiful for its lines and colours and not for its mythological connexion with Iris, or Noah's rainbow.
     From: report of Johann Herbart (works [1830]) by Leo Tolstoy - What is Art? Ch.3
     A reaction: A nice counterexample to Tolstoy's own theory. The example is one of a natural beauty, but it would be harder to find examples in human art. How much the artist may feel, though, has little to do with the success of a work of art.
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
6017 Nomos is king [Pindar]
     Full Idea: Nomos is king.
     From: Pindar (poems [c.478 BCE], S 169), quoted by Thomas Nagel - The Philosophical Culture
     A reaction: This seems to be the earliest recorded shot in the nomos-physis wars (the debate among sophists about moral relativism). It sounds as if it carries the full relativist burden - that all that matters is what has been locally decreed.