9 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) |
| 16755 | The possible Aristotelian view that forms are real and active principles is clearly wrong [Fine,K, by Pasnau] |
| Full Idea: Aristotle seems to have a possible basis for the belief [in individual forms], namely that forms are real and active principles in the world, which is denied by any right-minded modern. | |
| From: report of Kit Fine (A Puzzle Concerning Matter and Form [1994], p.19) by Robert Pasnau - Metaphysical Themes 1274-1671 24.3 n8 | |
| A reaction: Pasnau says this is the view of forms promoted by the scholastics, whereas Aristotle's own view should be understood as 'metaphysical'. |
| 20645 | Heat is a state of vibration, not a substance [Joule] |
| Full Idea: We consider heat not as a substance but as a state of vibration. | |
| From: James Joule (works [1870]), quoted by Peter Watson - Convergence 01 'Nature's' | |
| A reaction: The puzzle is that giving accurate accounts of vibrations, heat and movement require a quantitative substance, energy. But all we have here is movement, and the denial of a substance. Energy is 'nature's currency system'. |
| 20972 | Joule showed that energy converts to heat, and heat to energy [Joule, by Papineau] |
| Full Idea: James Joule established the equivalence of heat and mechanical energy, in the sense of showing that a specific amount of heat will always be produced by the expenditure of a given amount of energy, and vice versa. | |
| From: report of James Joule (works [1870]) by David Papineau - Thinking about Consciousness App 4.2 | |
| A reaction: This was a major step towards the law of conservation of energy. |