### All the ideas for Michael Burke, Yale Kamisar and John Mayberry

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41 ideas

###### 2. Reason / D. Definition / 2. Aims of Definition
 17774 Definitions make our intuitions mathematically useful [Mayberry]
###### 2. Reason / E. Argument / 6. Conclusive Proof
 17773 Proof shows that it is true, but also why it must be true [Mayberry]
###### 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 [Mayberry]
 17796 There is a semi-categorical axiomatisation of set-theory [Mayberry]
###### 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 [Mayberry]
###### 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 [Mayberry]
###### 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 [Mayberry]
###### 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 [Mayberry]
###### 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 [Mayberry]
###### 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
 17791 Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
###### 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 [Mayberry]
###### 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
 17790 No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
###### 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
 17778 Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
 17780 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
 17779 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
###### 5. Theory of Logic / K. Features of Logics / 6. Compactness
 17789 No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
###### 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 [Mayberry]
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
 17781 Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
 17782 Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
 17799 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
 17797 Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
###### 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
 17775 If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
 17776 The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
 17777 Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
 17804 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
###### 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 [Mayberry]
###### 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 [Mayberry]
###### 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 [Mayberry]
 17802 We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
 17805 Set theory is not just another axiomatised part of mathematics [Mayberry]
###### 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 [Mayberry]
###### 9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
 16235 Persistence conditions cannot contradict, so there must be a 'dominant sortal' [Burke,M, by Hawley]
 14753 The 'dominant' of two coinciding sortals is the one that entails the widest range of properties [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
 16072 'The rock' either refers to an object, or to a collection of parts, or to some stuff [Burke,M, by Wasserman]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / b. Cat and its tail
 14751 Tib goes out of existence when the tail is lost, because Tib was never the 'cat' [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
 13278 Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki]
 16234 Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley]
 16071 Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / d. Coincident objects
 14750 Two entities can coincide as one, but only one of them (the dominant sortal) fixes persistence conditions [Burke,M, by Sider]
###### 24. Applied Ethics / C. Death Issues / 5. Euthanasia
 4050 We only allow voluntary euthanasia to someone who is both sane and crazed by pain [Kamisar]
 4051 People will volunteer for euthanasia because they think other people want them dead [Kamisar]