Ideas from 'Naturalism in Mathematics' by Penelope Maddy [1997], by Theme Structure
[found in 'Naturalism in Mathematics' by Maddy,Penelope [OUP 2000,0198250754]].
green numbers give full details 
back to texts

expand these ideas
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
18195

A Large Cardinal Axiom would assert everincreasing stages in the hierarchy

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
18191

Axiom of Infinity: completed infinite collections can be treated mathematically

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 / p. Axiom of Reducibility
18169

Axiom of Reducibility: propositional functions are extensionally predicative

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

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

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 / i. Cardinal infinity
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)

18196

An 'inaccessible' cardinal cannot be reached by union sets or power 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
18184

Making set theory foundational to mathematics leads to very fruitful axioms

18185

Unified set theory gives a final court of appeal for mathematics

18183

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

18186

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

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

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
18207

Maybe applications of continuum mathematics are all idealisations

18204

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

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 / 11. 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
