11 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) |
| 17809 | Gödel showed that the syntactic approach to the infinite is of limited value [Kreisel] |
| Full Idea: Usually Gödel's incompleteness theorems are taken as showing a limitation on the syntactic approach to an understanding of the concept of infinity. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 05) |
| 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] |
| 17810 | The study of mathematical foundations needs new non-mathematical concepts [Kreisel] |
| Full Idea: It is necessary to use non-mathematical concepts, i.e. concepts lacking the precision which permit mathematical manipulation, for a significant approach to foundations. We currently have no concepts of this kind which we can take seriously. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 06) | |
| A reaction: Music to the ears of any philosopher of mathematics, because it means they are not yet out of a job. |
| 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) |
| 3291 | Emergent properties appear at high levels of complexity, but aren't explainable by the lower levels [Nagel] |
| Full Idea: The supposition that a diamond or organism should truly have emergent properties is that they appear at certain complex levels of organisation, but are not explainable (even in principle) in terms of any more fundamental properties of the system. | |
| From: Thomas Nagel (Panpsychism [1979], p.186) |
| 3290 | Given the nature of heat and of water, it is literally impossible for water not to boil at the right heat [Nagel] |
| Full Idea: Given what heat is and what water is, it is literally impossible for water to be heated beyond a certain point at normal atmospheric pressure without boiling. | |
| From: Thomas Nagel (Panpsychism [1979], p.186) |
| 17811 | The natural conception of points ducks the problem of naming or constructing each point [Kreisel] |
| Full Idea: In analysis, the most natural conception of a point ignores the matter of naming the point, i.e. how the real number is represented or by what constructions the point is reached from given points. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 13) | |
| A reaction: This problem has bothered me. There are formal ways of constructing real numbers, but they don't seem to result in a name for each one. |