Ideas from 'On the Question of Absolute Undecidability' by Peter Koellner [2006], by Theme Structure

[found in 'Philosophia Mathematica' (ed/tr -) [- ,]].

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
Mathematical set theory has many plausible stopping points, such as finitism, and predicativism
'Reflection principles' say the whole truth about sets can't be captured
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
We have no argument to show a statement is absolutely undecidable
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / j. Large cardinals
There are at least eleven types of large cardinal, of increasing logical strength
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
PA is consistent as far as we can accept, and we expand axioms to overcome limitations
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
Arithmetical undecidability is always settled at the next stage up