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) |
| 19369 | Lull's combinatorial art would articulate all the basic concepts, then show how they combine [Lull, by Arthur,R] |
| Full Idea: Lull proposed a combinatorial art. He wanted to reconcile Islam and Christianity by articulating the basic concepts that their belief systems held in common, and then inventing a device that would allow these concepts to be combined. | |
| From: report of Ramon (Ars Magna [1305]) by Richard T.W. Arthur - Leibniz 2 Intro | |
| A reaction: Leibniz's Universal Characteristic was an attempt at continuing Lull's project. Lull's plan rested on Aristotle's categories. |
| 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) |
| 20890 | Why do sexual relationships need permanence, if other relationships don't? [Punzo] |
| Full Idea: What is the reason for demanding permanence in the relationship of sexual partners when we do not see such permanence as being importance to other human relationships? | |
| From: Vincent C. Punzo (Morality and Human Sexuality [1969], p.220) | |
| A reaction: The distinction may not be that simple. 'Loyalty' must certainly be mentioned. Friends can legitimately drift apart, but to desert a close friend at a time of great need might be as great a crime as adultery. When is loyalty particularly needed? |
| 20891 | Does engaging in sexual intercourse really need no more thought than playing tennis? [Punzo] |
| Full Idea: It seems strange for a man and a woman to give no more thought to the question of whether they should engage in sexual intercourse than to the question of whether they shoud play tennis. | |
| From: Vincent C. Punzo (Morality and Human Sexuality [1969], p.221) | |
| A reaction: This strikes me as a reasonable point, but times have moved on since 1969, and for plenty of people nowadays playing tennis is a bigger issue than having sex, because of the time, equipment and effort involved. |
| 19371 | Nine principles of God: goodness, greatness, eternity, power, wisdom, will, virtue, truth and glory [Lull, by Arthur,R] |
| Full Idea: Lull restricted himself to only nine 'absolute principles' of God: goodness, greatness, eternity, power, wisdom, will, virtue, truth and glory | |
| From: report of Ramon (Ars Magna [1305]) by Richard T.W. Arthur - Leibniz 2 'Combinatorics' | |
| A reaction: Leibniz responded that God's perfections are infinite in number, and thus beyond human comprehension. Lull cut them down to nine, because he was designing a sort of conceptual logic that employed them. |