Combining Philosophers
Ideas for Benjamin Constant, Kurt Gödel and Diogenes (Sin)
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19 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 / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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10046
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The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel]
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10868
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The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg]
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13517
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If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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17885
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Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]
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10614
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The real reason for Incompleteness in arithmetic is inability to define truth in a language [Gödel]
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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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3198
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Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey]
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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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10039
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Some arithmetical problems require assumptions which transcend arithmetic [Gödel]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
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10043
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Mathematical objects are as essential as physical objects are for perception [Gödel]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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10271
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Basic mathematics is related to abstract elements of our empirical ideas [Gödel]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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10045
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Impredicative definitions are admitted into ordinary mathematics [Gödel]
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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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