Ideas from 'Defending the Axioms' by Penelope Maddy [2011], by Theme Structure

[found in 'Defending the Axioms' by Maddy,Penelope [OUP 2013,978-0-19-967148-9]].

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

17610

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

5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism

17620

Critics of if-thenism say that not all starting points, even consistent ones, are worth studying

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

17625

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

17605

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

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

17618

Set-theory tracks the contours of mathematical depth and fruitfulness

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