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) |
| 22244 | 'Partial reference' is when the subject thinks two objects are one object [Field,H, by Recanati] |
| Full Idea: A subject's thought is about A, but, unbeknownst to the subject, B is substituted for A. Then there is Field's 'partial reference', because the subject's thought is still partially about A, even though they are following B. | |
| From: report of Hartry Field (Theory Change and the Indeterminacy of Reference [1973]) by François Recanati - Mental Files in Flux 2 | |
| A reaction: Used to interpret a well-known case: Wally says of Udo 'he needs a haircut'; Zach looks at someone else and says 'he sure does'. Recanati explains it by mental files. |
| 22474 | Unlike aesthetic evaluation, moral evaluation needs a concept of responsibility [Foot] |
| Full Idea: Moral, as opposed to aesthetic, evaluation does require some distinction between actions for which we are responsible and those for which we are not responsible. | |
| From: Philippa Foot (Nietzsche's Immoralism [1991], p.154) | |
| A reaction: It is hard to disagree with this, but difficult to give a precise account of responsibility, probably because it is not an all-or-nothing matter. If we accept responsibility for our controlled actions, why not for our considered aesthetic judgements? |
| 22472 | The practice of justice may well need a recognition of human equality [Foot] |
| Full Idea: I wonder whether the practice of justice may not absolutely require a certain recognition of equality between human beings, not a pretence of the equality of talents, but something deeper. | |
| From: Philippa Foot (Nietzsche's Immoralism [1991], p.152) | |
| A reaction: {My 'something deeper' is expressed by Foot in a quotation from Gertrude Stein]. This may well be the most fundamental division which runs across a society - between those who accept and those reject human equality. |