Ideas of Penelope Maddy, by Theme
[American, b.1950, Professor of Logic and Philosophy of Science at the University of California, Irvine.]
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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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13011
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New axioms are being sought, to determine the size of the continuum
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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 / b. Axiom of Extensionality I
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13013
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The Axiom of Extensionality seems to be analytic
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13014
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Extensional sets are clearer, simpler, unique and expressive
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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13021
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The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
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13022
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Infinite sets are essential for giving an account of the real numbers
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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 / g. Axiom of Powers VI
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13023
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The Power Set Axiom is needed for, and supported by, accounts of the continuum
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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 / j. Axiom of Choice IX
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13024
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Efforts to prove the Axiom of Choice have failed
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13025
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Modern views say the Choice set exists, even if it can't be constructed
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13026
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A large array of theorems depend on the Axiom of Choice
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17610
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The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
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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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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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13019
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The Iterative Conception says everything appears at a stage, derived from the preceding appearances
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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13018
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Limitation of Size is a vague intuition that over-large sets may generate paradoxes
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4. Formal Logic / F. Set Theory ST / 7. Natural Sets
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17824
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The master science is physical objects divided into sets
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8755
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Maddy replaces pure sets with just objects and perceived sets of objects [Shapiro]
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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10594
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Henkin semantics is more plausible for plural logic than for second-order logic
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5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
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17620
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Critics of if-thenism say that not all starting points, even consistent ones, are worth studying
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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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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
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17605
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Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
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17625
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If two mathematical themes coincide, that suggest a single deep truth
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
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18190
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Completed infinities resulted from giving foundations to calculus
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18171
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Cantor and Dedekind brought completed infinities into mathematics
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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17615
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Every infinite set of reals is either countable or of the same size as the full set of reals
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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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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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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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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17825
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Set theory (unlike the Peano postulates) can explain why multiplication is commutative
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17826
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Standardly, numbers are said to be sets, which is neat ontology and epistemology
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17828
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Numbers are properties of sets, just as lengths are properties of physical objects
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10718
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A natural number is a property of sets [Oliver]
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18184
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Making set theory foundational to mathematics leads to very fruitful axioms
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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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18183
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Set theory brings mathematics into one arena, where interrelations become clearer
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18185
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Unified set theory gives a final court of appeal for mathematics
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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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18163
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Mathematics rests on the logic of proofs, and on the set theoretic axioms
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17618
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Set-theory tracks the contours of mathematical depth and fruitfulness
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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17827
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Sets exist where their elements are, but numbers are more like universals
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17830
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Number theory doesn't 'reduce' to set theory, because sets have number properties
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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17823
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If mathematical objects exist, how can we know them, and which objects are they?
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6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
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8756
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Intuition doesn't support much mathematics, and we should question its reliability [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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17733
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We know mind-independent mathematical truths through sets, which rest on experience [Jenkins]
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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 / 4. Mathematical Empiricism / c. Against mathematical empiricism
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17614
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The connection of arithmetic to perception has been idealised away in modern infinitary mathematics
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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17829
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Number words are unusual as adjectives; we don't say 'is five', and numbers always come first
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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 / 10. 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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