Ideas from 'On Formally Undecidable Propositions' by Kurt Gödel [1931], by Theme Structure
[found in 'From Frege to Gödel 1879-1931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0-674-32449-8]].
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3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
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21752
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Prior to Gödel we thought truth in mathematics consisted in provability [Quine]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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17835
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Gödel show that the incompleteness of set theory was a necessity [Hallett,M]
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
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17886
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The limitations of axiomatisation were revealed by the incompleteness theorems [Koellner]
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5. Theory of Logic / K. Features of Logics / 2. Consistency
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10071
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Second Incompleteness: nice theories can't prove their own consistency [Smith,P]
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5. Theory of Logic / K. Features of Logics / 3. Soundness
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19123
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If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Halbach/Leigh]
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5. Theory of Logic / K. Features of Logics / 5. Incompleteness
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10621
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Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P]
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17888
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The undecidable sentence can be decided at a 'higher' level in the system
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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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10132
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There can be no single consistent theory from which all mathematical truths can be derived [George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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10072
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First Incompleteness: arithmetic must always be incomplete [Smith,P]
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3198
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Gödel showed that arithmetic is either incomplete or inconsistent [Rey]
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9590
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Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Nagel/Newman]
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11069
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Gödel's Second says that semantic consequence outruns provability [Hanna]
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10118
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First Incompleteness: a decent consistent system is syntactically incomplete [George/Velleman]
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10122
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Second Incompleteness: a decent consistent system can't prove its own consistency [George/Velleman]
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10611
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There is a sentence which a theory can show is true iff it is unprovable [Smith,P]
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10867
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'This system can't prove this statement' makes it unprovable either way [Clegg]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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8747
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Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Shapiro]
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17. Mind and Body / C. Functionalism / 2. Machine Functionalism
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3192
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Basic logic can be done by syntax, with no semantics [Rey]
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