Combining Texts

All the ideas for 'works', 'Actualism and Thisness' 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)
5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
10121 Contradiction is not a sign of falsity, nor lack of contradiction a sign of truth [Pascal]
     Full Idea: Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.
     From: Blaise Pascal (works [1660]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: [Quoted in Auden and Kronenberger's Book of Aphorisms] Presumably we would now say that contradiction is a purely formal, syntactic notion, and not a semantic one. If you hit a contradiction, something has certainly gone wrong.
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)
9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity
16463 Adams says actual things have haecceities, but not things that only might exist [Adams,RM, by Stalnaker]
     Full Idea: Adams favours haecceitism about actual things but no haecceities for things that might exist but don't.
     From: report of Robert Merrihew Adams (Actualism and Thisness [1981]) by Robert C. Stalnaker - Mere Possibilities 4.2
     A reaction: This contrasts with Plantinga, who proposes necessary essences for everything, even for what might exist. Plantinga sounds crazy to me, Adams merely interesting but not too plausible.