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
Definitions make our intuitions mathematically useful
2. Reason / E. Argument / 6. Conclusive Proof
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
Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
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
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
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
Limitation of size is part of the very conception of a set
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
The mainstream of modern logic sees it as a branch of mathematics
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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
Only second-order logic can capture mathematical structure up to isomorphism
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
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. L÷wenheim-Skolem Theorems
No L÷wenheim-Skolem logic can axiomatise real analysis
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
'Classificatory' axioms aim at revealing similarity in morphology of structures
Axiomatiation relies on isomorphic structures being essentially the same
'Eliminatory' axioms get rid of traditional ideal and abstract objects
5. Theory of Logic / K. Features of Logics / 6. Compactness
No logic which can axiomatise arithmetic can be compact or complete
6. Mathematics / A. Nature of Mathematics / 2. Quantity
Greek quantities were concrete, and ratio and proportion were their science
Real numbers were invented, as objects, to simplify and generalise 'quantity'
6. Mathematics / A. Nature of Mathematics / 3. Numbers / d. Natural numbers
Cantor presented the totality of natural numbers as finite, not infinite
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers can be eliminated, by axiom systems for complete ordered fields
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
Cantor extended the finite (rather than 'taming the infinite')
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
If proof and definition are central, then mathematics needs and possesses foundations
The ultimate principles and concepts of mathematics are presumed, or grasped directly
Foundations need concepts, definition rules, premises, and proof rules
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
1st-order PA is only interesting because of results which use 2nd-order PA
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
It is only 2nd-order isomorphism which suggested first-order PA completeness
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
We don't translate mathematics into set theory, because it comes embodied in that way
Set theory is not just another axiomatised part of mathematics
Set theory is not just first-order ZF, because that is inadequate for mathematics
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Real numbers as abstracted objects are now treated as complete ordered fields