### Ideas from 'Understanding the Infinite' by Shaughan Lavine [1994], by Theme Structure

#### [found in 'Understanding the Infinite' by Lavine,Shaughan [Harvard 1994,0-674-92117-8]].

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###### 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
###### 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
###### 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
###### 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
###### 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
###### 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
###### 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
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
 15898 The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules
 15920 Pure collections of things obey Choice, but collections defined by a rule may not
###### 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
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
 15932 The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs
 15900 The iterative conception of set wasn't suggested until 1947
 15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate
###### 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
 15943 Limitation of Size is not self-evident, and seems too strong
###### 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
###### 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
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 15934 Mathematical proof by contradiction needs the law of excluded middle
###### 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
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
 15942 Every rational number, unlike every natural number, is divisible by some other number
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
 15922 For the real numbers to form a set, we need the Continuum Hypothesis to be true
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / h. Reals from Cauchy
 18250 Cauchy gave a necessary condition for the convergence of a sequence
 18251 Irrational numbers are the limits of Cauchy sequences of rational numbers
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
 15904 The two sides of the Cut are, roughly, the bounding commensurable ratios
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
 15912 Counting results in well-ordering, and well-ordering makes counting possible
###### 6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
 15947 The infinite is extrapolation from the experience of indefinitely large size
 15949 The theory of infinity must rest on our inability to distinguish between very large sizes
###### 6. Mathematics / A. Nature of Mathematics / 4. The Infinite / c. Potential infinite
 15940 The intuitionist endorses only the potential infinite
###### 6. Mathematics / A. Nature of Mathematics / 4. The Infinite / f. Uncountable infinities
 15909 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal
###### 6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
 15917 Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal
 15915 Ordinals are basic to Cantor's transfinite, to count the sets
 15893 Cantor's theory concerns collections which can be counted, using the ordinals
###### 6. Mathematics / A. Nature of Mathematics / 4. The Infinite / i. Cardinal infinity
 15918 Paradox: there is no largest cardinal, but the class of everything seems to be the largest
###### 6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
 15929 Set theory will found all of mathematics - except for the notion of proof
###### 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'
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 15928 Intuitionism rejects set-theory to found mathematics