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) |
| 13074 | Only natural kinds and their members have real essences [Suárez, by Cover/O'Leary-Hawthorne] |
| Full Idea: On Suarez's account, only natural kinds and their members have real essences. | |
| From: report of Francisco Suárez (works [1588]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 1.3.1 n21 | |
| A reaction: Interesting. Rather than say that everything is a member of some kind, we leave quirky individuals out, with no essence at all. What is the status of the very first exemplar of a given kind? |
| 22725 | When players don't meet again, defection is the best strategy [Axelrod] |
| Full Idea: When players will never meet again, the strategy of defection is the only stable strategy. | |
| From: Robert Axelrod (The Evolution of Co-Operation [1984], 5) | |
| A reaction: This gives good grounds for any community's mistrust of transient strangers, such as tourists. And yet any sensible tourist will want communities to trust tourists, and will therefore behave in a reliable way. |
| 22724 | Good strategies avoid conflict, respond to hostility, forgive, and are clear [Axelrod] |
| Full Idea: Successful game strategies avoid unnecessary conflict, are provoked by an uncalled for defection, forgive after a provocation, and behave clearly so the other player can adapt. | |
| From: Robert Axelrod (The Evolution of Co-Operation [1984], 1) | |
| A reaction: [compressed] Exactly what you would expect from a nice but successful school teacher. The strategies for success in these games is the same as the rules for educating a person into cooperative behaviour. TIT FOR TAT does all these. |
| 7563 | The old 'influx' view of causation says it is a flow of accidental properties from A to B [Suárez, by Jolley] |
| Full Idea: The 'influx' model of causation says that causes involve a process of contagion, as it were; when the kettle boils, the gas infects the water inside the kettle with its own 'individual accident' of heat, which literally flows from one to the other. | |
| From: report of Francisco Suárez (works [1588]) by Nicholas Jolley - Leibniz Ch.2 | |
| A reaction: This nicely captures the scholastic target of Hume's sceptical thinking on the subject. However, see Idea 2542, where the idea of influx has had a revival. It is hard to see how the water could change if it didn't 'catch' something from the gas. |