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) |
| 12354 | A 'categorial' property is had by virtue of being or having an item from a category [Wedin] |
| Full Idea: A 'categorial' property is a property something has by virtue of being or having an item from one of the categories. | |
| From: Michael V. Wedin (Aristotle's Theory of Substance [2000], V.5) | |
| A reaction: I deny that these are 'properties'. A thing is categorised according to its properties. To denote the category as a further property is the route to madness (well, to a regress). |
| 12358 | Substance is a principle and a kind of cause [Wedin] |
| Full Idea: Substance [ousia] is a principle [arché] and a kind of cause [aitia]. | |
| From: Michael V. Wedin (Aristotle's Theory of Substance [2000], 1041a09) | |
| A reaction: The fact that substance is a cause is also the reason why substance is the ultimate explanation. It is here that I take the word 'power' to capture best what Aristotle has in mind. |
| 12346 | Form explains why some matter is of a certain kind, and that is explanatory bedrock [Wedin] |
| Full Idea: The form of a thing (of a given kind) explains why certain matter constitutes a thing of that kind, and with this, Aristotle holds, we have reached explanatory bedrock. | |
| From: Michael V. Wedin (Aristotle's Theory of Substance [2000], Intro) | |
| A reaction: We must explain an individual tiger which is unusually docile. It must have an individual form which makes it a tiger, but also an individual form which makes it docile. |
| 1451 | Design is seen in the way ideas match the world, in the mechanisms of evolution, and in values [Tennant,FR, by PG] |
| Full Idea: There is evidence for design in the correspondence of pure ideas to the world, in the origin and mechanism of evolution, and in the existence of moral values and beauty. | |
| From: report of F.R. Tennant (Philosophical Theology [1930], II.IV) by PG - Db (ideas) |