14 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) |
| 13831 | Logic is based on transitions between sentences [Prawitz] |
| Full Idea: I agree entirely with Dummett that the right way to answer the question 'what is logic?' is to consider transitions between sentences. | |
| From: Dag Prawitz (Gentzen's Analysis of First-Order Proofs [1974], §04) | |
| A reaction: I always protest at this point that reliance on sentences is speciesism against animals, who are thereby debarred from reasoning. See the wonderful Idea 1875 of Chrysippus. Hacking's basic suggestion seems right. Transition between thoughts. |
| 13825 | Natural deduction introduction rules may represent 'definitions' of logical connectives [Prawitz] |
| Full Idea: With Gentzen's natural deduction, we may say that the introductions represent, as it were, the 'definitions' of the logical constants. The introductions are not literally understood as 'definitions'. | |
| From: Dag Prawitz (Gentzen's Analysis of First-Order Proofs [1974], 2.2.2) | |
| A reaction: [Hacking, in 'What is Logic? §9' says Gentzen had the idea that his rules actually define the constants; not sure if Prawitz and Hacking are disagreeing] |
| 13823 | In natural deduction, inferences are atomic steps involving just one logical constant [Prawitz] |
| Full Idea: In Gentzen's natural deduction, the inferences are broken down into atomic steps in such a way that each step involves only one logical constant. The steps are the introduction or elimination of the logical constants. | |
| From: Dag Prawitz (Gentzen's Analysis of First-Order Proofs [1974], 1.1) |
| 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) |
| 22971 | In moving from capitalism to communism a revolutionary dictatorship of the proletariat is needed [Marx] |
| Full Idea: Between the capitalist and communist society lies the revolutionary transformation of the one into the other. Corresponding to this is a political transition period in which the state can be nothing but the revolutionary dictatorship of the proletariat. | |
| From: Karl Marx (Critique of the Gotha Program [1875], IV) | |
| A reaction: This hugely influential idea was catastrophic for the twentieth century, because the leaders of the proletarian dictatorship adored and abused the power, and wouldn't give it up for some feeble next stage. |
| 22970 | Freedom is making the state subordinate to its society [Marx] |
| Full Idea: Freedom consists in converting the state from an organ superimposed on society into one completely subordinate to it. | |
| From: Karl Marx (Critique of the Gotha Program [1875], IV) | |
| A reaction: The intermediate stage is dictatorship of the proletariat (presumably exercised by the communist leadership). No twentieth century marxist state ever got near the freedom which Marx was seeking. A liberal society might achieve it! |
| 22969 | People who only have their labour power are the slaves of those permitting them to work [Marx] |
| Full Idea: The man who possesses no other property than his labour power must, in all conditions of society and culture, be the slave of other men who have made themselves the owners of the material conditions of labour. He can only work with their permission. | |
| From: Karl Marx (Critique of the Gotha Program [1875], I) | |
| A reaction: In a world of vast multinationals, the person giving the permission to work is nearly always dependent on some higher level permission. In any sort of society people can only work with the consensus of other people. |
| 22000 | From each according to his ability, to each according to his need [Marx] |
| Full Idea: From each according to his ability, to each according to his need. | |
| From: Karl Marx (Critique of the Gotha Program [1875]), quoted by Peter Singer - Marx 9 | |
| A reaction: Singer says this was not original to Marx, and he placed little emphasis on it. The obvious capitalist response is to ask how you will motivate someone who has huge abilities but few needs. It implies huge inequalities of altruism. |
| 22972 | Bourgeois 'freedom of conscience' just tolerates all sorts of religious intolerance [Marx] |
| Full Idea: Bourgeois 'freedom of conscience' is just the toleration of all possible kinds of religious unfreedom of conscience, and the workers' party should endeavour to liberate the conscience from the witchery of religion. | |
| From: Karl Marx (Critique of the Gotha Program [1875], IV) | |
| A reaction: We see this in modern 'faith' schools in the UK, which do not seem to be required to live up to the standards of freedom of belief expected in the rest of a liberal society. |