8 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) |
| 22329 | Logic is highly general truths abstracted from reality [Russell, by Glock] |
| Full Idea: In 1911 Russell held that the propositions of logic are supremely general truths about the most pervasive traits of reality, to which we have access by abstraction from non-logical propositions. | |
| From: report of Bertrand Russell (Philosophical Implications of Mathematical logic [1911]) by Hans-Johann Glock - What is Analytic Philosophy? 2.4 | |
| A reaction: Glock says the rival views were Mill's inductions, psychologism, and Frege's platonism. Wittgenstein converted Russell to a fifth view, that logic is empty tautologies. I remain resolutely attached to Russell's abstraction view. |
| 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) |
| 16007 | I assume existence, rather than reasoning towards it [Kierkegaard] |
| Full Idea: I always reason from existence, not towards existence. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.40) | |
| A reaction: Kierkegaard's important premise to help show that theistic proofs for God's existence don't actually prove existence, but develop the content of a conception. [SY] |
| 16013 | Nothing necessary can come into existence, since it already 'is' [Kierkegaard] |
| Full Idea: Can the necessary come into existence? That is a change, and everything that comes into existence demonstrates that it is not necessary. The necessary already 'is'. | |
| From: Søren Kierkegaard (Philosophical Fragments [1844], p.74) | |
| A reaction: [SY] |
| 21569 | It is good to generalise truths as much as possible [Russell] |
| Full Idea: It is a good thing to generalise any truth as much as possible. | |
| From: Bertrand Russell (Philosophical Implications of Mathematical logic [1911], p.289) | |
| A reaction: An interesting claim, which seems to have a similar status to Ockham's Razor. Its best justification is pragmatic, and concerns strategies for coping with a big messy world. Russell's defence is in 'as much as possible'. |