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'reports', 'Understanding the Infinite' and 'Lecture on Nominalism'
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13 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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
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
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
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
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
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
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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15947
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The infinite is extrapolation from the experience of indefinitely large size [Lavine]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
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
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
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
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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