Ideas of John Mayberry, by Theme

[American, fl. 1994, Professor of Mathematics at the University of the Pacific, California.]

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2. Reason / D. Definition / 2. Aims of Definition
17774 Definitions make our intuitions mathematically useful
2. Reason / E. Argument / 6. Conclusive Proof
17773 Proof shows that it is true, but also why it must be true
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17795 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
17796 There is a semi-categorical axiomatisation of set-theory
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
17800 The misnamed Axiom of Infinity says the natural numbers are finite in size
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
17801 The set hierarchy doesn't rely on the dubious notion of 'generating' them
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
17803 Limitation of size is part of the very conception of a set
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
17786 The mainstream of modern logic sees it as a branch of mathematics
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
17788 First-order logic only has its main theorems because it is so weak
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
17791 Only second-order logic can capture mathematical structure up to isomorphism
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
17787 Big logic has one fixed domain, but standard logic has a domain for each interpretation
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17790 No Löwenheim-Skolem logic can axiomatise real analysis
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17778 Axiomatiation relies on isomorphic structures being essentially the same
17779 'Classificatory' axioms aim at revealing similarity in morphology of structures
17780 'Eliminatory' axioms get rid of traditional ideal and abstract objects
5. Theory of Logic / K. Features of Logics / 6. Compactness
17789 No logic which can axiomatise arithmetic can be compact or complete
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17784 Real numbers can be eliminated, by axiom systems for complete ordered fields
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
17782 Greek quantities were concrete, and ratio and proportion were their science
17781 Real numbers were invented, as objects, to simplify and generalise 'quantity'
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17797 Cantor extended the finite (rather than 'taming the infinite')
17799 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17775 If proof and definition are central, then mathematics needs and possesses foundations
17776 The ultimate principles and concepts of mathematics are presumed, or grasped directly
17777 Foundations need concepts, definition rules, premises, and proof rules
17804 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17792 1st-order PA is only interesting because of results which use 2nd-order PA
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793 It is only 2nd-order isomorphism which suggested first-order PA completeness
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794 Set theory is not just first-order ZF, because that is inadequate for mathematics
17802 We don't translate mathematics into set theory, because it comes embodied in that way
17805 Set theory is not just another axiomatised part of mathematics
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
17785 Real numbers as abstracted objects are now treated as complete ordered fields