### All the ideas for Kenelm Digby, Stephen Wolfram and Shaughan Lavine

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37 ideas

###### 4. Formal Logic / F. Set Theory ST / 1. Set Theory
 15945 Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
 15914 An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
 15921 Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
 15937 Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
 15936 The Power Set is just the collection of functions from one collection to another [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
 15899 Replacement was immediately accepted, despite having very few implications [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
 15930 Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
 15920 Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
 15898 The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
 15919 The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
 15900 The iterative conception of set wasn't suggested until 1947 [Lavine]
 15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
 15932 The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
 15933 Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
###### 4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
 15913 A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
###### 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
 15926 Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 15934 Mathematical proof by contradiction needs the law of excluded middle [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 15907 Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
 15942 Every rational number, unlike every natural number, is divisible by some other number [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 15922 For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
 18250 Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
 15904 The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
 15912 Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 20660 At one level maths and nature are very similar, suggesting some deeper origin [Wolfram]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
 15947 The infinite is extrapolation from the experience of indefinitely large size [Lavine]
 15949 The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
 15940 The intuitionist endorses only the potential infinite [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
 15909 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
 15915 Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
 15917 Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
 15918 Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 15929 Set theory will found all of mathematics - except for the notion of proof [Lavine]
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
 15935 Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 15928 Intuitionism rejects set-theory to found mathematics [Lavine]
###### 9. Objects / C. Structure of Objects / 4. Quantity of an Object
 16597 Quantity is the capacity to be divided [Digby]
###### 26. Natural Theory / A. Speculations on Nature / 7. Later Matter Theories / b. Corpuscles
 16731 Colours arise from the rarity, density and mixture of matter [Digby]
###### 27. Natural Reality / C. Space / 4. Substantival Space
 20659 Space and its contents seem to be one stuff - so space is the only existing thing [Wolfram]