9 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) |
| 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) |
| 13437 | A CAR and its major PART can become identical, yet seem to have different properties [Gallois] |
| Full Idea: At t1 there is a whole CAR, and a PART of it, which is everything except the right front wheel. At t2 the wheel is removed, leaving just PART, so that CAR is now PART. But PART was a proper part of CAR, and CAR had the front wheel. Different properties! | |
| From: André Gallois (Occasions of Identity [1998], 1.II) | |
| A reaction: [compressed summary] The problem is generated by appealing to Leibniz's Law. My immediate reaction is that this is the sort of trouble you get into if you include such temporal truths about things as 'properties'. |
| 16233 | Gallois hoped to clarify identity through time, but seems to make talk of it impossible [Hawley on Gallois] |
| Full Idea: A problem for Gallois is that he leaves us no way to talk about questions of genuine identity through time, and thus undercuts one motivation for his own position. | |
| From: comment on André Gallois (Occasions of Identity [1998]) by Katherine Hawley - How Things Persist 5.8 | |
| A reaction: Gallois seems to need a second theory of identity to support his Occasional Identity theory. Two things need an identity each, before we can say that the two identities coincide. (Time to read Gallois!) |
| 14755 | Gallois is committed to identity with respect to times, and denial of simple identity [Gallois, by Sider] |
| Full Idea: Gallois's core claim is that the identity relation holds with respect to times, ...and he must claim that there is no such thing as the relation of identity simpliciter. | |
| From: report of André Gallois (Occasions of Identity [1998]) by Theodore Sider - Four Dimensionalism 5.5 | |
| A reaction: Gallois is essentially responding to the statue and clay problem, but it seems a bit drastic to entirely change our concept of two things being identical, such as Hesperus and Phosphorus. 'Identity' seems to have several meanings; let's sort them out. |
| 16231 | Occasional Identity: two objects can be identical at one time, and different at others [Gallois, by Hawley] |
| Full Idea: Gallois' Occasional Identity Thesis is that objects can be identical at one time without being identical at all times. | |
| From: report of André Gallois (Occasions of Identity [1998]) by Katherine Hawley - How Things Persist 5.4 | |
| A reaction: The analogy is presumably with two crossing roads being identical at one place but not at others. It is a major misunderstanding to infer from Special Relativity that time is just like space. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |