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
'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
A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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
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
Axiom of Reducibility: propositional functions are extensionally predicative
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
'Propositional functions' are propositions with a variable as subject or predicate
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite
Cantor and Dedekind brought completed infinities into mathematics
Completed infinities resulted from giving foundations to calculus
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / i. Cardinal infinity
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / j. Large cardinals
For any cardinal there is always a larger one (so there is no set of all sets)
An 'inaccessible' cardinal cannot be reached by union sets or power sets
Infinity has degrees, and large cardinals are the heart of set theory
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / m. Limits
Theorems about limits could only be proved once the real numbers were understood
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
The extension of concepts is not important to me
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
Frege solves the Caesar problem by explicitly defining each number
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / f. Zermelo numbers
For Zermelo the successor of n is {n} (rather than n U {n})
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / g. Von Neumann numbers
For Von Neumann the successor of n is n U {n} (rather than {n})
Von Neumann numbers are preferred, because they continue into the transfinite
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Mathematics rests on the logic of proofs, and on the set theoretic axioms
Unified set theory gives a final court of appeal for mathematics
Identifying geometric points with real numbers revealed the power of set theory
Making set theory foundational to mathematics leads to very fruitful axioms
Set theory brings mathematics into one arena, where interrelations become clearer
The line of rationals has gaps, but set theory provided an ordered continuum
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Maybe applications of continuum mathematics are all idealisations
Scientists posit as few entities as possible, but set theorist posit as many as possible
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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
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
Science idealises the earth's surface, the oceans, continuities, and liquids