10 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) |
| 14305 | In the truth-functional account a burnt-up match was soluble because it never entered water [Carnap] |
| Full Idea: If a wooden match was completely burned up yesterday, and never placed in water at any time, is it not the case, therefore, that the match is soluble (in the truth-functional view). This follows just from the antecedent being false. | |
| From: Rudolph Carnap (Testability and Meaning [1937], I.440), quoted by Stephen Mumford - Dispositions | |
| A reaction: This, along with Edgington's nice example of the conditional command (Idea ) seems conclusive against the truth-functional account. The only defence possible is some sort of pragmatic account about implicature. |
| 23104 | Dworkin believed we should promote equality, to increase autonomy [Dworkin, by Kekes] |
| Full Idea: Egalitarians believe that most often it is by promoting equality that autonomy is increased; this is the egalitarianism of such liberals as Ronald Dworkin. | |
| From: report of Ronald Dworkin (Taking Rights Seriously [1977]) by John Kekes - Against Liberalism 05.1 | |
| A reaction: Not my idea of equality. The whole point is to ascribe reasonable equality to everyone, including those with a limited capacity for autonomy. Equality is a consequence of universal respect. |
| 23257 | We can treat people as equals, or actually treat them equally [Dworkin, by Grayling] |
| Full Idea: Dworkin distinguishes between treating people as equals, that is, 'with equal concern and respect', and treating them equally. This latter can be unjust. | |
| From: report of Ronald Dworkin (Taking Rights Seriously [1977]) by A.C. Grayling - The Good State 2 | |
| A reaction: The big difference I see between them is that the first is mere words, and the second is actions. Cf. 'thoughts and prayers' after US school shootings. How about equal entitlements, all things being equal? |
| 18621 | Treating people as equals is the one basic value of all plausible political theories [Dworkin, by Kymlicka] |
| Full Idea: Dworkin suggests that every plausible political theory has the same ultimate value, which is equality - in the more abstract and fundamental sense of treating people 'as equals'. | |
| From: report of Ronald Dworkin (Taking Rights Seriously [1977], 179-83) by Will Kymlicka - Contemporary Political Philosophy (1st edn) | |
| A reaction: I associate this idea with Kant (who says they are equal by virtue of their rationality), so that's a pretty influential idea. I would associate the main challenge to this with Nietzsche. |