Combining Philosophers
Ideas for Archimedes, Stewart Shapiro and Fabrice Correia
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10 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10201
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Virtually all of mathematics can be modeled in set theory [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
13641
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Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
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8763
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
13676
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Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13677
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Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10213
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Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
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18243
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Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
18249
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Cauchy gave a formal definition of a converging sequence. [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18245
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Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
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