Combining Philosophers
Ideas for Hermarchus, Michael Dummett and Stephen Read
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7 ideas
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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10979
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Although second-order arithmetic is incomplete, it can fully model normal arithmetic [Read]
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10980
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Second-order arithmetic covers all properties, ensuring categoricity [Read]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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10554
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Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
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10997
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Von Neumann numbers are helpful, but don't correctly describe numbers [Read]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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9829
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The identity of a number may be fixed by something outside structure - by counting [Dummett]
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9828
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Numbers aren't fixed by position in a structure; it won't tell you whether to start with 0 or 1 [Dummett]
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9192
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The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett]
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