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Ideas of Michael Hallett, by Text

[British, fl. 1996, Professor at McGill University, Montreal.]

1996 Introduction to Zermelo's 1930 paper
p.1213 p.1213 The first-order ZF axiomatisation is highly non-categorical
     Full Idea: The first-order Sermelo-Fraenkel axiomatisation is highly non-categorical.
     From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1213)
p.1215 p.1215 Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal
     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).
p.1217 p.1217 Zermelo allows ur-elements, to enable the widespread application of set-theory
     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)
p.1217 p.1217 The General Continuum Hypothesis and its negation are both consistent with ZF
     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)