11 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) |
| 7301 | The phenomenalist says that to be is to be perceivable [Cardinal/Hayward/Jones] |
| Full Idea: Where the idealist says that to be (i.e. to exist) is to be perceived, the phenomenalist says that to be is to be perceivable. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: This is a nice phenomenalist slogan to add to Mill's well known one (Idea 3583). Expressed in this form, it looks false to me. What about neutrinoes? They weren't at all perceivable until recently. Maybe some physical stuff can never be perceived. |
| 7302 | Linguistic phenomenalism says we can eliminate talk of physical objects [Cardinal/Hayward/Jones] |
| Full Idea: Linguistic phenomenalism argues that it is possible to remove all talk of physical objects from our speech with no loss of meaning. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: I find this proposal unappealing. My basic objection is that I cannot understand why anyone would refuse to even contemplate the question of WHY I am having a given group of consistent experiences, of (say) a table kind. |
| 7303 | If we lack enough sense-data, are we to say that parts of reality are 'indeterminate'? [Cardinal/Hayward/Jones] |
| Full Idea: The problem with taking sense-data as basic is that some data can appear indeterminate. If we can't discern the colour of someone's eyes, or the number of sides of a complex figure, are we to say that there is no fact about those things? | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: I like that. How many electrons are there in the sun? Such things cannot just be reduced to talk of sense-data, as there is obviously a vast gap between the data and the facts. As usual, ontology and epistemology must be kept separate. |
| 7299 | Primary qualities can be described mathematically, unlike secondary qualities [Cardinal/Hayward/Jones] |
| Full Idea: All the primary qualities lend themselves readily to mathematical or geometric description. ...but it seems that secondary qualities are less amenable to being represented mathematically. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: As a believer in the primary/secondary distinction, I welcome this point. This is either evidence for the external reality of primary qualities, or an interesting observation about maths. Do we make the primary/secondary distinction because we do maths? |
| 7300 | An object cannot remain an object without its primary qualities [Cardinal/Hayward/Jones] |
| Full Idea: An object cannot lack shape, size, position or motion and remain an object. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: This points towards the essentialist view (see Idea 5453). This does raise the question of whether an object could lose its colour with impugnity, or the quality of sound that it makes when struck. |
| 7297 | My justifications might be very coherent, but totally unconnected to the world [Cardinal/Hayward/Jones] |
| Full Idea: My beliefs could be well justified in coherentist terms, while not accurately representing the world, and my system of beliefs could be completely free-floating. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.3) | |
| A reaction: This nicely encapsulates to correspondence objection to coherence theory. One thing missing from the coherence account is that beliefs aren't chosen for their coherence, but are mostly unthinkingly triggered by experiences. |
| 1497 | For Anaximenes nature is air, which takes different forms by rarefaction and condensation [Anaximenes, by Simplicius] |
| Full Idea: Unlike Anaximander, Anaximenes' underlying nature is not boundless, but specific, since he says that it is air, and claims that it is thanks to rarefaction and condensation that it manifests in different forms in different things. | |
| From: report of Anaximenes (fragments/reports [c.546 BCE], A5) by Simplicius - On Aristotle's 'Physics' 9.24.26- |