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'fragments/reports', 'Investigations into Logical Deduction' and 'On Formally Undecidable Propositions'
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10 ideas
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 [Gödel, by George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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3198
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Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
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10072
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First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P]
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9590
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Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman]
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11069
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Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna]
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10118
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First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman]
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10122
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Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by 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 [Gödel, by Smith,P]
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10867
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'This system can't prove this statement' makes it unprovable either way [Gödel, by 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 [Gödel, by Shapiro]
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