10 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) |
| 13554 | True greatness is never allowing events to disturb you [Seneca] |
| Full Idea: There is no more reliable proof of greatness than to be in a state where nothing can happen to make you disturbed. | |
| From: Seneca the Younger (On Anger (Book 3) [c.60], §06) | |
| A reaction: He specifically opposes Aristotle's view that there are times when anger is appropriate, and failure to be very angry indeed is a failure of character. |
| 13556 | Every night I critically review how I have behaved during the day [Seneca] |
| Full Idea: When the lamp has been removed from my sight, and my wife, no stranger now to my habit, has fallen silent, I examine the whole of my day and retrace my actions and words; I hide nothing from myself. | |
| From: Seneca the Younger (On Anger (Book 3) [c.60], §36) |
| 13552 | Anger is an extreme vice, threatening sanity, and gripping whole states [Seneca] |
| Full Idea: Other vices drive the mind on, anger hurls it headlong; ..other vices revolt from good sense, this one from sanity; ...other vices seize individuals, this is the one passion that sometimes takes hold of an entire state. | |
| From: Seneca the Younger (On Anger (Book 3) [c.60], §01) | |
| A reaction: He particularly dislikes anger because it is the vice that leads to violence. |
| 13553 | Anger is a vice which afflicts good men as well as bad [Seneca] |
| Full Idea: Other vile passions affect only the worst sort of men, but anger creeps up even on enlightened me who are otherwise sane. | |
| From: Seneca the Younger (On Anger (Book 3) [c.60], §04) | |
| A reaction: A very interesting observation for anyone who is trying to analyse the key issues in virtue theory. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |