Ideas from 'What Required for Foundation for Maths?' by John Mayberry [1994], by Theme Structure
[found in 'Philosophy of Mathematics: anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21870-x]].
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expand these ideas
2. Reason / D. Definition / 2. Aims of Definition
17774
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Definitions make our intuitions mathematically useful
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2. Reason / E. Argument / 6. Conclusive Proof
17773
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Proof shows that it is true, but also why it must be true
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17795
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Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
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17796
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There is a semi-categorical axiomatisation of set-theory
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
17800
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The misnamed Axiom of Infinity says the natural numbers are finite in size
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
17801
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The set hierarchy doesn't rely on the dubious notion of 'generating' them
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
17803
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Limitation of size is part of the very conception of a set
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5. Theory of Logic / A. Overview of Logic / 2. History of Logic
17786
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The mainstream of modern logic sees it as a branch of mathematics
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
17788
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First-order logic only has its main theorems because it is so weak
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
17791
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Only second-order logic can capture mathematical structure up to isomorphism
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5. Theory of Logic / G. Quantification / 2. Domain of Quantification
17787
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Big logic has one fixed domain, but standard logic has a domain for each interpretation
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17790
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No Löwenheim-Skolem logic can axiomatise real analysis
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17778
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Axiomatiation relies on isomorphic structures being essentially the same
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17779
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'Classificatory' axioms aim at revealing similarity in morphology of structures
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17780
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'Eliminatory' axioms get rid of traditional ideal and abstract objects
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5. Theory of Logic / K. Features of Logics / 6. Compactness
17789
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No logic which can axiomatise arithmetic can be compact or complete
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17784
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Real numbers can be eliminated, by axiom systems for complete ordered fields
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
17782
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Greek quantities were concrete, and ratio and proportion were their science
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17781
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Real numbers were invented, as objects, to simplify and generalise 'quantity'
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
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17797
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Cantor extended the finite (rather than 'taming the infinite')
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6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17775
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If proof and definition are central, then mathematics needs and possesses foundations
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly
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17777
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Foundations need concepts, definition rules, premises, and proof rules
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17792
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1st-order PA is only interesting because of results which use 2nd-order PA
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way
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17805
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Set theory is not just another axiomatised part of mathematics
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9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
17785
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Real numbers as abstracted objects are now treated as complete ordered fields
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