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) |
13766 | 'If' is the same as 'given that', so the degrees of belief should conform to probability theory [Ramsey, by Ramsey] |
Full Idea: Ramsey suggested that 'if', 'given that' and 'on the supposition that' come to the same thing, and that the degrees of belief in the antecedent should then conform to probability theory. | |
From: report of Frank P. Ramsey (Truth and Probability [1926]) by Frank P. Ramsey - Law and Causality B | |
A reaction: [compressed] |
8825 | It seems impossible to logically deduce physical knowledge from indubitable sense data [Kim] |
Full Idea: It is agreed on all hands that the classical epistemological project, conceived as one of deductively validating physical knowledge from indubitable sensory data, cannot succeed. | |
From: Jaegwon Kim (What is 'naturalized epistemology'? [1988], p.304) | |
A reaction: This is the 'Enlightenment Project', which had a parallel in morality. Kim refers to the difficulty as 'The Humean Predicament'. Hume also hoped that induction might be deductive. One obvious move is to expand from 'deduction' to 'reason'. |
19143 | Ramsey gave axioms for an uncertain agent to decide their preferences [Ramsey, by Davidson] |
Full Idea: Ramsey gave an axiomatic treatment of preference in the face of uncertainty, when applied to a particular agent. | |
From: report of Frank P. Ramsey (Truth and Probability [1926]) by Donald Davidson - Truth and Predication 2 | |
A reaction: This is evidently the beginnings of Bayesian decision theory. |