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
18194

'Forcing' can produce new models of ZFC from old models

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
13011

New axioms are being sought, to determine the size of the continuum

18195

A Large Cardinal Axiom would assert everincreasing stages in the hierarchy

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
13013

The Axiom of Extensionality seems to be analytic

13014

Extensional sets are clearer, simpler, unique and expressive

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
13021

The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics

13022

Infinite sets are essential for giving an account of the real numbers

18191

Axiom of Infinity: completed infinite collections can be treated mathematically

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
13023

The Power Set Axiom is needed for, and supported by, accounts of the continuum

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
18193

The Axiom of Foundation says every set exists at a level in the set hierarchy

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13024

Efforts to prove the Axiom of Choice have failed

13025

Modern views say the Choice set exists, even if it can't be constructed

13026

A large array of theorems depend on the Axiom of Choice

17610

The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
18169

Axiom of Reducibility: propositional functions are extensionally predicative

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
13019

The Iterative Conception says everything appears at a stage, derived from the preceding appearances

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
13018

Limitation of Size is a vague intuition that overlarge sets may generate paradoxes

4. Formal Logic / F. Set Theory ST / 7. Natural Sets
17824

The master science is physical objects divided into sets

8755

Maddy replaces pure sets with just objects and perceived sets of objects [Shapiro]

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
10594

Henkin semantics is more plausible for plural logic than for secondorder logic

5. Theory of Logic / C. Ontology of Logic / 3. IfThenism
17620

Critics of ifthenism say that not all starting points, even consistent ones, are worth studying

5. Theory of Logic / E. Structures of Logic / 1. Logical Form
18168

'Propositional functions' are propositions with a variable as subject or predicate

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17605

Hilbert's geometry and Dedekind's real numbers were role models for axiomatization

17625

If two mathematical themes coincide, that suggest a single deep truth

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
18190

Completed infinities resulted from giving foundations to calculus

18171

Cantor and Dedekind brought completed infinities into mathematics

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17615

Every infinite set of reals is either countable or of the same size as the full set of reals

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18196

An 'inaccessible' cardinal cannot be reached by union sets or power sets

18172

Infinity has degrees, and large cardinals are the heart of set theory

18175

For any cardinal there is always a larger one (so there is no set of all sets)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18187

Theorems about limits could only be proved once the real numbers were understood

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
18182

The extension of concepts is not important to me

18177

In the ZFC hierarchy it is impossible to form Frege's set of all threeelement sets

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18164

Frege solves the Caesar problem by explicitly defining each number

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17825

Set theory (unlike the Peano postulates) can explain why multiplication is commutative

17826

Standardly, numbers are said to be sets, which is neat ontology and epistemology

17828

Numbers are properties of sets, just as lengths are properties of physical objects

10718

A natural number is a property of sets [Oliver]

18184

Making set theory foundational to mathematics leads to very fruitful axioms

18186

Identifying geometric points with real numbers revealed the power of set theory

18183

Set theory brings mathematics into one arena, where interrelations become clearer

18185

Unified set theory gives a final court of appeal for mathematics

18188

The line of rationals has gaps, but set theory provided an ordered continuum

18163

Mathematics rests on the logic of proofs, and on the set theoretic axioms

17618

Settheory tracks the contours of mathematical depth and fruitfulness

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
17827

Sets exist where their elements are, but numbers are more like universals

17830

Number theory doesn't 'reduce' to set theory, because sets have number properties

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
17823

If mathematical objects exist, how can we know them, and which objects are they?

6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
8756

Intuition doesn't support much mathematics, and we should question its reliability [Shapiro]

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17733

We know mindindependent mathematical truths through sets, which rest on experience [Jenkins]

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
18204

Scientists posit as few entities as possible, but set theorist posit as many as possible

18207

Maybe applications of continuum mathematics are all idealisations

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17614

The connection of arithmetic to perception has been idealised away in modern infinitary mathematics

6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
17829

Number words are unusual as adjectives; we don't say 'is five', and numbers always come first

6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neologicism
18167

We can get arithmetic directly from HP; Law V was used to get HP from the definition of number

7. Existence / D. Theories of Reality / 10. Ontological Commitment / e. Ontological commitment problems
18205

The theoretical indispensability of atoms did not at first convince scientists that they were real

15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
18206

Science idealises the earth's surface, the oceans, continuities, and liquids
