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
17884

Mathematical set theory has many plausible stopping points, such as finitism, and predicativism

17893

'Reflection principles' say the whole truth about sets can't be captured

5. Theory of Logic / K. Features of Logics / 5. Incompleteness
17894

We have no argument to show a statement is absolutely undecidable

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

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

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
