Ideas from 'The Philosophy of Mathematics' by Michael Dummett [1998], by Theme Structure

[found in 'Philosophy 2: further through the subject' (ed/tr Grayling,A.C.) [OUP 1998,0-19-875178-8]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality
The main alternative to ZF is one which includes looser classes as well as sets
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Intuitionists reject excluded middle, not for a third value, but for possibility of proof
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
First-order logic concerns objects; second-order adds properties, kinds, relations and functions
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths and inference are characterized either syntactically or semantically
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Ordinals seem more basic than cardinals, since we count objects in sequence
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
The number 4 has different positions in the naturals and the wholes, with the same structure