6 ideas
| 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). |
| 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) |
| 19406 | I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz] |
| Full Idea: I am so much for the actual infinite that instead of admitting that nature abhors it, as is commonly said, I hold that it affects nature everywhere in order to indicate the perfections of its author. | |
| From: Gottfried Leibniz (Reply to Foucher [1693], p.99) | |
| A reaction: I would have thought that, for Leibniz, while infinities indicate the perfections of their author, that is not the reason why they exist. God wasn't, presumably, showing off. Leibniz does not think we can actually know these infinities. |
| 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) |
| 14963 | Surely the past phases of a thing are not parts of the thing? [Broad] |
| Full Idea: It is plainly contrary to common sense to say that the phases in the history of a thing are parts of the thing. | |
| From: C.D. Broad (Examination of McTaggart's Philosophy [1933], I.349-50), quoted by Richard Cartwright - Scattered Objects n18 | |
| A reaction: Nicely expressed! To suggest that me ten years ago is a mere part of some huge me, or that you are only talking to a part of me now, is a very long way indeed from normal usage. |