display all the ideas for this combination of texts
8 ideas
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) |
18153 | A number is a repeated operation [Wittgenstein] |
Full Idea: A number is the index of an operation. | |
From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.021) | |
A reaction: Roughly, this means that a number indicates how many times some basic operation has been performed. Bostock 2009:286 expounds the idea. |
18160 | The concept of number is just what all numbers have in common [Wittgenstein] |
Full Idea: The concept of number is simply what is common to all numbers, the general form of number. The concept of number is the variable number. | |
From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.022) |
18161 | The theory of classes is superfluous in mathematics [Wittgenstein] |
Full Idea: The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not accidental generality. | |
From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.031) | |
A reaction: This fits Russell's no-class theory, which rests everything instead on propositional functions. |
6849 | Wittgenstein hated logicism, and described it as a cancerous growth [Wittgenstein, by Monk] |
Full Idea: Wittgenstein didn't just have an arguments against logicism; he hated logicism, and described is as a cancerous growth. | |
From: report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by Ray Monk - Interview with Baggini and Stangroom p.12 | |
A reaction: This appears to have been part of an inexplicable personal antipathy towards Russell. Wittgenstein appears to have developed a dislike of all reductionist ideas in philosophy. |
23509 | The logic of the world is shown by tautologies in logic, and by equations in mathematics [Wittgenstein] |
Full Idea: The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. | |
From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.22) | |
A reaction: White observes that this is Wittgenstein distinguishing logic from mathematics, and thus distancing himself from logicism. But see T 6.2. |