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'fragments/reports', 'What Required for Foundation for Maths?' and 'works'
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18 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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8717
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Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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17784
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Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
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17782
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Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
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17781
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Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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17775
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If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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17792
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1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
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17805
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Set theory is not just another axiomatised part of mathematics [Mayberry]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
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10113
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The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
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10115
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Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman]
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6. Mathematics / C. Sources of Mathematics / 8. Finitism
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10116
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Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman]
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