17833 | 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) |
17834 | 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). |
17837 | 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) |
17836 | 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) |