Combining Texts

All the ideas for 'Defending the Axioms', 'The Source of Necessity' and 'Principles of Theoretical Logic'

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13 ideas

4. Formal Logic / C. Predicate Calculus PC / 1. Predicate Calculus PC
17750 The first clear proof of the consistency of the first order predicate logic was in 1928 [Hilbert/Ackermann, by Walicki]
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 [Maddy]
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 [Maddy]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17605 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
17625 If two mathematical themes coincide, that suggest a single deep truth [Maddy]
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 [Maddy]
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 [Maddy]
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 [Maddy]
10. Modality / C. Sources of Modality / 1. Sources of Necessity
12432 Explanation of necessity must rest on something necessary or something contingent [Hale]
12434 Why is this necessary, and what is necessity in general; why is this necessary truth true, and why necessary? [Hale]
12435 The explanation of a necessity can be by a truth (which may only happen to be a necessary truth) [Hale]
10. Modality / C. Sources of Modality / 3. Necessity by Convention
12433 If necessity rests on linguistic conventions, those are contingent, so there is no necessity [Hale]
10. Modality / C. Sources of Modality / 4. Necessity from Concepts
12436 Concept-identities explain how we know necessities, not why they are necessary [Hale]