Ideas from 'Naturalism in Mathematics' by Penelope Maddy [1997], by Theme Structure
[found in 'Naturalism in Mathematics' by Maddy,Penelope [OUP 2000,0-19-825075-4]].
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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18194
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'Forcing' can produce new models of ZFC from old models
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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18195
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A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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18191
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Axiom of Infinity: completed infinite collections can be treated mathematically
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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18193
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The Axiom of Foundation says every set exists at a level in the set hierarchy
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
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18169
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Axiom of Reducibility: propositional functions are extensionally predicative
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5. Theory of Logic / E. Structures of Logic / 1. Logical Form
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18168
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'Propositional functions' are propositions with a variable as subject or predicate
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
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18171
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Cantor and Dedekind brought completed infinities into mathematics
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18190
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Completed infinities resulted from giving foundations to calculus
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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18175
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For any cardinal there is always a larger one (so there is no set of all sets)
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18196
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An 'inaccessible' cardinal cannot be reached by union sets or power sets
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18172
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Infinity has degrees, and large cardinals are the heart of set theory
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
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18187
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Theorems about limits could only be proved once the real numbers were understood
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
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18182
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The extension of concepts is not important to me
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18177
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In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
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18164
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Frege solves the Caesar problem by explicitly defining each number
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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18163
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Mathematics rests on the logic of proofs, and on the set theoretic axioms
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18185
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Unified set theory gives a final court of appeal for mathematics
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18183
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Set theory brings mathematics into one arena, where interrelations become clearer
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18186
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Identifying geometric points with real numbers revealed the power of set theory
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18184
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Making set theory foundational to mathematics leads to very fruitful axioms
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18188
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The line of rationals has gaps, but set theory provided an ordered continuum
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
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18204
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Scientists posit as few entities as possible, but set theorist posit as many as possible
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18207
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Maybe applications of continuum mathematics are all idealisations
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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18167
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We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
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7. Existence / D. Theories of Reality / 11. Ontological Commitment / e. Ontological commitment problems
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18205
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The theoretical indispensability of atoms did not at first convince scientists that they were real
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15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
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18206
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Science idealises the earth's surface, the oceans, continuities, and liquids
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