Combining Texts
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'Extrinsic Properties', 'Against the Professors (six books)' and 'Foundations without Foundationalism'
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9 ideas
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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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
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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
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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 / 5. The Infinite / g. Continuum Hypothesis
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
13657
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First-order arithmetic can't even represent basic number theory [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13656
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Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
13663
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Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
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