Combining Texts
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'Difference and Repetition', 'The Confessions' and 'Understanding the Infinite'
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16 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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15907
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Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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15942
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Every rational number, unlike every natural number, is divisible by some other number [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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15922
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For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
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18250
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Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
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15904
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The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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15912
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Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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15947
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The infinite is extrapolation from the experience of indefinitely large size [Lavine]
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15949
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The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
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15940
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The intuitionist endorses only the potential infinite [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
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15909
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'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
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15915
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Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
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15917
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Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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15918
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Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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15929
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Set theory will found all of mathematics - except for the notion of proof [Lavine]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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15935
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Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
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15928
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Intuitionism rejects set-theory to found mathematics [Lavine]
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