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) |
| 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) |
| 18948 | There is an object for every set of properties (some of which exist, and others don't) [Parsons,T, by Sawyer] |
| Full Idea: According to Terence Parsons, there is an object corresponding to every set of properties. To some of those sets of properties there corresponds an object that exists, and to others there corresponds an object that does not exist (a nonexistent object). | |
| From: report of Terence Parsons (Nonexistent Objects [1980]) by Sarah Sawyer - Empty Names 5 | |
| A reaction: This I take to be the main source of the modern revival of Meinong's notorious view of objects (attacked by Russell). I always find the thought 'a round square is square' to be true, and in need of a truthmaker. But must a round square be non-triangular? |
| 3626 | Knowing the attributes is enough to reveal a substance [Descartes] |
| Full Idea: I have never thought that anything more is required to reveal a substance than its various attributes. | |
| From: René Descartes (Reply to Fifth Objections [1641], 360) |
| 3630 | Our thinking about external things doesn't disprove the existence of innate ideas [Descartes] |
| Full Idea: You can't prove that Praxiteles never made any statues on the grounds that he did not get from within himself the marble from which he sculpted them. | |
| From: René Descartes (Reply to Fifth Objections [1641], 362) |
| 3631 | A blind man may still contain the idea of colour [Descartes] |
| Full Idea: How do you know that there is no idea of colour in a man born blind? | |
| From: René Descartes (Reply to Fifth Objections [1641], 363) |
| 3640 | Possible existence is a perfection in the idea of a triangle [Descartes] |
| Full Idea: Possible existence is a perfection in the idea of a triangle, just as necessary existence is a perfection in the idea of God. | |
| From: René Descartes (Reply to Fifth Objections [1641], 383) |
| 3639 | Necessary existence is a property which is uniquely part of God's essence [Descartes] |
| Full Idea: In the case of God necessary existence is in fact a property in the strictest sense of the term, since it applies to him alone and forms a part of his essence as it does of no other thing | |
| From: René Descartes (Reply to Fifth Objections [1641], 383) |