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) |
| 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) |
| 18086 | Weierstrass eliminated talk of infinitesimals [Weierstrass, by Kitcher] |
| Full Idea: Weierstrass effectively eliminated the infinitesimalist language of his predecessors. | |
| From: report of Karl Weierstrass (works [1855]) by Philip Kitcher - The Nature of Mathematical Knowledge 10.6 |
| 18092 | Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock] |
| Full Idea: After Weierstrass had stressed the importance of limits, one now needed to be able to prove the existence of such limits. | |
| From: report of Karl Weierstrass (works [1855]) by David Bostock - Philosophy of Mathematics 4.4 | |
| A reaction: The solution to this is found in work on series (going back to Cauchy), and on Dedekind's cuts. |
| 1757 | The Electra: she knows this man, but not that he is her brother [Eucleides, by Diog. Laertius] |
| Full Idea: The 'Electra': Electra knows that Orestes is her brother, but not that this man is Orestes, so she knows and does not know her brother simultaneously. | |
| From: report of Eucleides (fragments/reports [c.410 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Eu.4 | |
| A reaction: Hence we distinguish 'know of', 'know that' and 'know how'. Hence Russell makes 'knowledge by acquaintance' fundamental, and descriptions come later. |
| 3028 | The chief good is unity, sometimes seen as prudence, or God, or intellect [Eucleides] |
| Full Idea: The chief good is unity, which is known by several names, for at one time people call it prudence, at another time God, at another intellect, and so on. | |
| From: Eucleides (fragments/reports [c.410 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 02.9.2 | |
| A reaction: So the chief good is what unites and focuses our moral actions. Kant calls that 'the will'. |