Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'A Completeness Theorem in Modal Logic' and 'Lectures 1930-32 (student notes)'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

5 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18738 We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein]
     Full Idea: We say we get nearer to root-2 by adding further figures after the decimal point: 1.1412.... This suggests there is something we can get nearer to, but the analogy is a false one.
     From: Ludwig Wittgenstein (Lectures 1930-32 (student notes) [1931], Notes)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
18708 Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein]
     Full Idea: 'Infinite' is not an answer to the question 'How many?', since the infinite is not a number. ...Infinity is the property of a law, not of an extension.
     From: Ludwig Wittgenstein (Lectures 1930-32 (student notes) [1931], A VII.2)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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)