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) |
| 7751 | Meaning needs an intention to induce a belief, and a recognition that this is the speaker's intention [Grice] |
| Full Idea: For a statement to have (non-naturally) meant something, not merely must it have been 'uttered' with the intention of inducing a certain belief, but also the utterer must have intended an 'audience' to recognise the intention behind the utterance. | |
| From: H. Paul Grice (Meaning [1957], p.43) | |
| A reaction: This is Grice's famous and distinctive theory of meaning. I am struck by the problem of a password, which seems to have a quite different intention from its literal meaning. Also a speaker with two different audiences and opposite intentions. |
| 7752 | Only the utterer's primary intention is relevant to the meaning [Grice] |
| Full Idea: Only what I may call the primary intention of an utterer is relevant to the (non-natural) meaning of an utterance. | |
| From: H. Paul Grice (Meaning [1957], p.47) | |
| A reaction: This sounds okay for simple statements, but gets really tricky with complex statements, such as very ironic remarks delivered to an audience of diverse people. |
| 7753 | We judge linguistic intentions rather as we judge non-linguistic intentions, so they are alike [Grice] |
| Full Idea: To show that the criteria for judging linguistic intentions are very like the criteria for judging non-linguistic intentions is to show that linguistic intentions are very like non-linguistic intentions. | |
| From: H. Paul Grice (Meaning [1957], p.48) | |
| A reaction: This hint at the end of his paper is one of the key attractions of Grice's view. It offers an account of language that fits it into the world of animal communication and evolution. It never seems to quite capture the way meaning goes beyond intentions. |
| 1515 | Pythagoreans believe it is absurd to seek for goodness anywhere except with the gods [Iamblichus] |
| Full Idea: The thinking behind Pythagorean philosophy is that people behave in an absurd fashion if they try to find any source for the good other than the gods. | |
| From: Iamblichus (Life of Pythagoras [c.290], 137) |