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) |
| 10121 | Contradiction is not a sign of falsity, nor lack of contradiction a sign of truth [Pascal] |
| Full Idea: Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth. | |
| From: Blaise Pascal (works [1660]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: [Quoted in Auden and Kronenberger's Book of Aphorisms] Presumably we would now say that contradiction is a purely formal, syntactic notion, and not a semantic one. If you hit a contradiction, something has certainly gone wrong. |
| 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) |
| 9086 | The idea of abstract objects is not ontological; it comes from the epistemological idea of abstraction [Plantinga] |
| Full Idea: The notion of an abstract object comes from the notion of abstraction; it is in origin an epistemological rather than an ontological category. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.232) | |
| A reaction: Etymology doesn't prove anything. However, if you define abstract objects as not existing in space or time, you must recognise that this may only be because that is how humans imaginatively created them in the first place. |
| 9087 | Theists may see abstract objects as really divine thoughts [Plantinga] |
| Full Idea: Theists may find attractive a view popular among medieval philosophers from Augustine on: that abstract objects are really divine thoughts. More exactly, propositions are divine thoughts, properties divine concepts, and sets divine collections. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.233) | |
| A reaction: Hm. I pass this on because we should be aware that there is a theological history to discussions of abstract objects, and some people have vested interests in keeping them outside of the natural world. Aren't properties natural? Does God gerrymander sets? |
| 9085 | If propositions are concrete they don't have to exist, and so they can't be necessary truths [Plantinga] |
| Full Idea: Someone who believes propositions are concrete cannot agree that some propositions are necessary. For propositions are contingent beings, and could have failed to exist. But if they fail to exist, then they fail to be true. | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.230) | |
| A reaction: [compressed] He implies the actual existence of an infinity of trivial, boring or ridiculous necessary truths. I suspect that he is just confusing a thought with its content. Or we might just treat necessary propositions as hypothetical. |
| 9084 | Propositions can't just be in brains, because 'there are no human beings' might be true [Plantinga] |
| Full Idea: If propositions are brain inscriptions, then if there had been no human beings there would have been no propositions. But then 'there are no human beings' would have been true, so there would have been at least one truth (and thus one proposition). | |
| From: Alvin Plantinga (Why Propositions cannot be concrete [1993], p.229) | |
| A reaction: This would make 'there are no x's' true for any value of x apart from actual objects, which implies an infinity of propositions. Does Plantinga really believe that these all exist? He may be confusing propositions with facts. |