12 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) |
| 13418 | The old problems with the axiom of choice are probably better ascribed to the law of excluded middle [Parsons,C] |
| Full Idea: The difficulties historically attributed to the axiom of choice are probably better ascribed to the law of excluded middle. | |
| From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], §2) | |
| A reaction: The law of excluded middle was a target for the intuitionists, so presumably the debate went off in that direction. |
| 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) |
| 13419 | If functions are transfinite objects, finitists can have no conception of them [Parsons,C] |
| Full Idea: The finitist may have no conception of function, because functions are transfinite objects. | |
| From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], §4) | |
| A reaction: He is offering a view of Tait's. Above my pay scale, but it sounds like a powerful objection to the finitist view. Maybe there is a finitist account of functions that could be given? |
| 13417 | If a mathematical structure is rejected from a physical theory, it retains its mathematical status [Parsons,C] |
| Full Idea: If experience shows that some aspect of the physical world fails to instantiate a certain mathematical structure, one will modify the theory by sustituting a different structure, while the original structure doesn't lose its status as part of mathematics. | |
| From: Charles Parsons (Review of Tait 'Provenance of Pure Reason' [2009], §2) | |
| A reaction: This seems to be a beautifully simple and powerful objection to the Quinean idea that mathematics somehow only gets its authority from physics. It looked like a daft view to begin with, of course. |
| 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) |