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) |
| 20062 | If a desire leads to a satisfactory result by an odd route, the causal theory looks wrong [Chisholm] |
| Full Idea: If someone wants to kill his uncle to inherit a fortune, and having this desire makes him so agitated that he loses control of his car and kills a pedestrian, who turns out to be his uncle, the conditions of the causal theory seem to be satisfied. | |
| From: Roderick Chisholm (Freedom and Action [1966]), quoted by Rowland Stout - Action 6 'Deviant' | |
| A reaction: This line of argument has undermined all sorts of causal theories that were fashionable in the 1960s and 70s. Explanation should lead to understanding, but a deviant causal chain doesn't explain the outcome. The causal theory can be tightened. |
| 20054 | There has to be a brain event which is not caused by another event, but by the agent [Chisholm] |
| Full Idea: There must be some event A, presumably some cerebral event, which is not caused by any other event, but by the agent. | |
| From: Roderick Chisholm (Freedom and Action [1966], p.20), quoted by Rowland Stout - Action 4 'Agent' | |
| A reaction: I'm afraid this thought strikes me as quaintly ridiculous. What kind of metaphysics can allow causation outside the natural nexus, yet occuring within the physical brain? This is a relic of religious dualism. Let it go. |
| 22481 | There is no restitution after a dilemma, if it only involved the agent, or just needed an explanation [Foot, by PG] |
| Full Idea: The 'remainder' after a dilemma can't be a matter of apology and restitution, because the dilemma may only involve the agent's own life, and in the case of broken promises we only owe an explanation, if the breaking is justifiable. | |
| From: report of Philippa Foot (Moral Dilemmas Revisited [1995], p.183) by PG - Db (ideas) | |
| A reaction: But what if someone has been financially ruined by it? If the agent feels guilty about that, is getting over it the rational thing to do? (Foot says that is an new obligation, and not part of the original dilemma). |
| 22482 | I can't understand how someone can be necessarily wrong whatever he does [Foot] |
| Full Idea: I do not see how …we can know how to interpret the idea of a situation in which someone will necessarily be wrong whatever he does. | |
| From: Philippa Foot (Moral Dilemmas Revisited [1995], p.188) | |
| A reaction: Seems right. If you think of hideous dilemmas (frequent in wartime), there must always be a right thing to do (or two equally right things to do), even if the outcome is fairly hideous. Just distinguish the right from the good. |