display all the ideas for this combination of texts
5 ideas
| 3907 | Could you be intellectually acquainted with numbers, but unable to count objects? [Scruton] |
| Full Idea: Could someone have a perfect intellectual acquaintance with numbers, but be incapable of counting a flock of sheep? | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.6) |
| 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) |
| 3908 | If maths contains unprovable truths, then maths cannot be reduced to a set of proofs [Scruton] |
| Full Idea: If there can be unprovable truths of mathematics, then mathematics cannot be reduced to the proofs whereby we construct it. | |
| From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.7) |