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
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17884
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Mathematical set theory has many plausible stopping points, such as finitism, and predicativism
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17893
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'Reflection principles' say the whole truth about sets can't be captured
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5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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17894
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We have no argument to show a statement is absolutely undecidable
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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17890
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There are at least eleven types of large cardinal, of increasing logical strength
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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17887
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PA is consistent as far as we can accept, and we expand axioms to overcome limitations
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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17891
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Arithmetical undecidability is always settled at the next stage up
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