8 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) |
10009 | Substitutional quantification is just a variant of Tarski's account [Wallace, by Baldwin] |
Full Idea: In a famous paper, Wallace argued that all interpretations of quantifiers (including the substitutional interpretation) are, in the end, variants of that proposed by Tarski (in 1936). | |
From: report of Wallace, J (On the Frame of Reference [1970]) by Thomas Baldwin - Interpretations of Quantifiers | |
A reaction: A significant-looking pointer. We must look elsewhere for Tarski's account, which will presumably subsume the objectual interpretation as well. The ontology of Tarski's account of truth is an enduring controversy. |
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) |
22818 | Liberals are not too individualistic, because people recognise and value social relations [Kymlicka] |
Full Idea: It is alleged that liberals fail to recognise that people are naturally social or communal. …But liberals believe that people form and join social relations in which they come to understand and pursue the good. | |
From: Will Kymlicka (Liberal Individualism and Liberal Neutrality [1989], Conc) | |
A reaction: This is particulary aimed at communitarians, who see liberalism as based on a distorted concept of people as isolated beings. Personally I am beginning to shift my views from Aristotelian communitarianism to modern liberalism, so I like this idea. |