8 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) |
| 6175 | External identification doesn't mean external location, as with sunburn [Davidson, by Rowlands] |
| Full Idea: Davidson observes that the inference from a thought being identified by a relation to something outside the head does not entail that the thought is not wholly in the head, just as sunburn is identified by external factors, but is still in the skin. | |
| From: report of Donald Davidson (Knowing One's Own Mind [1987]) by Mark Rowlands - Externalism Ch.8 | |
| A reaction: Rowlands (an externalist) agrees, and this strikes me as correct, and it needs to be one of the fixed points in any assessment of externalism. |
| 20082 | Bodily movements are not actions, which are really the tryings within bodily movement [Hornsby, by Stout,R] |
| Full Idea: Hornsby claims the basic description of action is in terms of trying, that all actions (even means of doing other actions) are actions of trying, and that tryings (and therefore actions) are interior to bodily movements (which are thus not essential). | |
| From: report of Jennifer Hornsby (Actions [1980]) by Rowland Stout - Action 9 'Trying' | |
| A reaction: [compression of his summary] There is no regress with explaining the 'action' of trying, because it is proposed that trying is the most basic thing in all actions. If you are paralysed, your trying does not result in action. Too mentalistic? |