Combining Texts
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'Metaphysics', 'On Formally Undecidable Propositions' and 'The Bacchae'
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12 ideas
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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 / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
17851
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Number is plurality measured by unity [Aristotle]
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17843
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The idea of 'one' is the foundation of number [Aristotle]
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17850
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Each many is just ones, and is measured by the one [Aristotle]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
9793
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Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
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