10 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) |
23420 | In a pluralist society we can't expect a community united around one conception of the good [Rawls] |
Full Idea: The fact of pluralism means that the hope of political community must be abandoned, if by such a community we mean a political society united in affirming a general and comprehensive conception of the good. | |
From: John Rawls (The Idea of Overlapping Consensus [1987]), quoted by Will Kymlicka - Community 'legitimacy' | |
A reaction: A moderate pluralism might be manageable, but strong, diverse and dogmatic beliefs among sub-groups probably make it impossible. |
21004 | Hart (against Bentham) says human rights are what motivate legal rights [Hart,HLA, by Sen] |
Full Idea: Whereas Bentham saw rights as a 'child of law', Herbert Hart's view takes the form of seeing human rights as, in effect, 'parents of law'; they motivate specific legislations. | |
From: report of H.L.A. Hart (The Concept of Law [1961]) by Amartya Sen - The Idea of Justice 17 'Ethics' | |
A reaction: [He cites Hart 1955 'Are there any natural rights?'] I agree with Hart. It is clearer if the parents of law are not referred to as 'rights'. You can demand a right, but it is only a right when it is awarded to you. |
20932 | Positive law needs secondary 'rules of recognition' for their correct application [Hart,HLA, by Zimmermann,J] |
Full Idea: Hart says we have secondary legal 'rules of recognition', by which primary positive law is recognised and applied in a regulated manner. | |
From: report of H.L.A. Hart (The Concept of Law [1961]) by Jens Zimmermann - Hermeneutics: a very short introduction 6 'Rules' | |
A reaction: The example of the authority of a particular court is given. |
20931 | Hart replaced positivism with the democratic requirement of the people's acceptance [Hart,HLA, by Zimmermann,J] |
Full Idea: Hart replaced Austin's concept of positive law as sovereign command with a more democratic ideal. In modern law-based societies the authority of law depends on the people's acceptance of a law's enduring validity. | |
From: report of H.L.A. Hart (The Concept of Law [1961]) by Jens Zimmermann - Hermeneutics: a very short introduction 6 'Hart' | |
A reaction: Presumably the ancestor of this view is the social contract of Hobbes and Locke. |