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Ideas of John Mayberry, by Text

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

1994 What Required for Foundation for Maths?
p.405-1 p.405 Definitions make our intuitions mathematically useful
p.405-1 p.405 If proof and definition are central, then mathematics needs and possesses foundations
p.405-1 p.405 The ultimate principles and concepts of mathematics are presumed, or grasped directly
p.405-2 p.405 Foundations need concepts, definition rules, premises, and proof rules
p.405-2 p.405 Proof shows that it is true, but also why it must be true
p.406-2 p.406 'Classificatory' axioms aim at revealing similarity in morphology of structures
p.406-2 p.406 Axiomatiation relies on isomorphic structures being essentially the same
p.407-1 p.407 'Eliminatory' axioms get rid of traditional ideal and abstract objects
p.407-2 p.407 Greek quantities were concrete, and ratio and proportion were their science
p.407-2 p.407 Real numbers were invented, as objects, to simplify and generalise 'quantity'
p.408-2 p.408 Real numbers can be eliminated, by axiom systems for complete ordered fields
p.408-2 p.408 Real numbers as abstracted objects are now treated as complete ordered fields
p.410-1 p.410 The mainstream of modern logic sees it as a branch of mathematics
p.410-2 p.410 Big logic has one fixed domain, but standard logic has a domain for each interpretation
p.411-2 p.411 First-order logic only has its main theorems because it is so weak
p.412-1 p.412 Only second-order logic can capture mathematical structure up to isomorphism
p.412-1 p.412 No L÷wenheim-Skolem logic can axiomatise real analysis
p.412-1 p.412 No logic which can axiomatise arithmetic can be compact or complete
p.412-1 p.412 1st-order PA is only interesting because of results which use 2nd-order PA
p.412-1 p.412 It is only 2nd-order isomorphism which suggested first-order PA completeness
p.412-2 p.412 Set theory is not just first-order ZF, because that is inadequate for mathematics
p.413-2 p.413 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
p.413-2 p.413 There is a semi-categorical axiomatisation of set-theory
p.414-2 p.414 The misnamed Axiom of Infinity says the natural numbers are finite in size
p.414-2 p.414 The set hierarchy doesn't rely on the dubious notion of 'generating' them
p.414-2 p.414 Cantor presented the totality of natural numbers as finite, not infinite
p.414-2 p.414 Cantor extended the finite (rather than 'taming the infinite')
p.414-2 p.414 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
p.415-1 p.415 We don't translate mathematics into set theory, because it comes embodied in that way
p.415-2 p.415 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
p.415-2 p.415 Limitation of size is part of the very conception of a set
p.416-1 p.416 Set theory is not just another axiomatised part of mathematics