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) |
| 16755 | The possible Aristotelian view that forms are real and active principles is clearly wrong [Fine,K, by Pasnau] |
| Full Idea: Aristotle seems to have a possible basis for the belief [in individual forms], namely that forms are real and active principles in the world, which is denied by any right-minded modern. | |
| From: report of Kit Fine (A Puzzle Concerning Matter and Form [1994], p.19) by Robert Pasnau - Metaphysical Themes 1274-1671 24.3 n8 | |
| A reaction: Pasnau says this is the view of forms promoted by the scholastics, whereas Aristotle's own view should be understood as 'metaphysical'. |
| 23829 | National leaders want to preserve necessary order - but always the existing order [Weil] |
| Full Idea: Those in command see their duty as defending order, without which no social life can survive; and the only order they conceive is the existing one. | |
| From: Simone Weil (The Power of Words [1934], p.249) | |
| A reaction: She sympathises with them, because a new order is such an unknown. But it always struck me as weird that traditions are preserved because they are traditions, and not because they are good. (My old school, for example!). |
| 23828 | National prestige consists of behaving as if you could beat the others in a war [Weil] |
| Full Idea: What is called national prestige consists in behaving always in such a way as to demoralise other nations by giving them the impression that, if it comes to war, one would certainly defeat them. | |
| From: Simone Weil (The Power of Words [1934], p.244) | |
| A reaction: It's true. No nation gains prestige because of the happy lives of its citizens, or the creativity of its culture. |
| 23827 | Modern wars are fought in the name of empty words which are given capital letters [Weil] |
| Full Idea: For our contemporaries the role of Helen in the Trojan War is is played by words with capital letters. …When empty words are given capital letters, then, on the slightest pretext, men will begin shedding blood for them and piling up ruin in their name. | |
| From: Simone Weil (The Power of Words [1934], p.241) | |
| A reaction: This seems particularly true of the 1930s, where specific dogmatic ideologies seemed to grip and divide people. Simple aggressive nationalism seems to be the cause of current wars, now the fear of Communism has diminished. |