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) |
| 7496 | Rules and duties are based on the will, as that is all we control [Montaigne] |
| Full Idea: Since actions and performances are not wholly in our power and since nothing is really in our power but our will - it is on the will that all the rules and duties of Man are based and established. | |
| From: Michel de Montaigne (I.7 Our deeds are judged by intention [1580], p.0028) | |
| A reaction: This is almost Kant's claim that the only truly good thing is a good will (e.g. Idea 3711). Aristotle disagrees, because a virtuous person should also have good desires. We may will to have good desires, but virtue requires actually having them. |
| 1470 | Belief in an afterlife may be unverifiable in this life, but it will be verifiable after death [Hick, by PG] |
| Full Idea: Religion is capable of 'eschatological verification', by reaching evidence at the end of life, even though falsification of its claims is never found in this life; a prediction of coming to a Celestial City must await the end of the journey. | |
| From: report of John Hick (Theology and Verification [1960], III) by PG - Db (ideas) |
| 1471 | It may be hard to verify that we have become immortal, but we could still then verify religious claims [Hick, by PG] |
| Full Idea: Verification of religious claims after death is only possible if the concept of surviving death is intelligible, and we can understand the concept of immortality, despite difficulties in being certain that we had reached it. | |
| From: report of John Hick (Theology and Verification [1960], IV) by PG - Db (ideas) |
| 1469 | Some things (e.g. a section of the expansion of PI) can be verified but not falsified [Hick, by PG] |
| Full Idea: Falsification and verification are not logically equivalent. For example, you might verify the claim that there will be three consecutive sevens in the infinite expansion of PI, but you could never falsify such a claim. | |
| From: report of John Hick (Theology and Verification [1960], §II) by PG - Db (ideas) |