9 ideas
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 16700 | In order to speak about time and successive entities, the 'present' must be enlarged [Wycliff] |
| Full Idea: It is clear from the way in which one must speak about time and other successive entities that talk about 'the present' must be enlarged. Otherwise it would have to be denied that any successive entity could exist, which is impossible. | |
| From: John Wycliff (De ente praedicamentali [1375], 20 p.189), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 18.3 | |
| A reaction: This is a lovely idea, even if it is not quite clear what it means. The mind seems to stretch out the now anyway (as the 'specious present'), so why not embrace that in language and conscious thought? |
| 16701 | To be successive a thing needs parts, which must therefore be lodged outside that instant [Wycliff] |
| Full Idea: If something is successive, it is successive with respect to its individual parts, which cannot exist at the same instant. Therefore it follows that many of its parts are lodged outside that instant. | |
| From: John Wycliff (De ente praedicamentali [1375], 20 p.189), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 18.3 | |
| A reaction: An obvious would be to say that there are therefore no successive entities, but Wycliff is appealing to our universal acceptance of them, and offering a transcendental argument. Nice move. |
| 15664 | Ideology is 'socially necessary illusion' or 'socially necessary false-consciousness' [Adorno, by Finlayson] |
| Full Idea: Adorno defines ideology as 'socially necessary illusion' or 'socially necessary false-consciousness' (and the young Habermas accepted something like this definition). | |
| From: report of Theodor W. Adorno (works [1955]) by James Gordon Finlayson - Habermas Ch.1:11 | |
| A reaction: The marxism seems to reside in the view that such things are always 'false'. If they gradually became 'true', would they cease to be ideology? Is it impossible for widespread beliefs to be 'true'? |