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Ideas of Shaughan Lavine, by Text

[American, fl. 2006, Professor at the University of Arizona.]

1994 Understanding the Infinite
2.5 p.33 Cauchy gave a necessary condition for the convergence of a sequence
4.2 p.92 Irrational numbers are the limits of Cauchy sequences of rational numbers
I p.3 Cantor's theory concerns collections which can be counted, using the ordinals
I p.4 The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules
I p.5 The iterative conception of set wasn't suggested until 1947
I p.5 Replacement was immediately accepted, despite having very few implications
II.6 p.38 The two sides of the Cut are, roughly, the bounding commensurable ratios
III.2 p.47 Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity
III.3 p.50 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal
III.4 p.53 Counting results in well-ordering, and well-ordering makes counting possible
III.4 p.53 An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one
III.4 p.53 A collection is 'well-ordered' if there is a least element, and all of its successors can be identified
III.4 p.54 Ordinals are basic to Cantor's transfinite, to count the sets
III.5 p.61 Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal
III.5 p.62 Paradox: there is no largest cardinal, but the class of everything seems to be the largest
IV.1 p.63 The 'logical' notion of class has some kind of definition or rule to characterise the class
IV.2 p.78 Pure collections of things obey Choice, but collections defined by a rule may not
IV.2 p.78 Collections of things can't be too big, but collections by a rule seem unlimited in size
IV.2 p.92 For the real numbers to form a set, we need the Continuum Hypothesis to be true
V.3 p.123 Second-order logic presupposes a set of relations already fixed by the first-order domain
V.3 p.133 Set theory will found all of mathematics - except for the notion of proof
V.3 n33 p.132 Intuitionism rejects set-theory to found mathematics
V.4 p.135 Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets
V.5 p.148 The iterative conception needs the Axiom of Infinity, to show how far we can iterate
V.5 p.149 The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs
V.5 p.150 Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement
VI.1 p.155 Mathematical proof by contradiction needs the law of excluded middle
VI.1 p.157 Modern mathematics works up to isomorphism, and doesn't care what things 'really are'
VI.1 p.160 The Power Set is just the collection of functions from one collection to another
VI.2 p.164 Those who reject infinite collections also want to reject the Axiom of Choice
VI.2 p.176 The intuitionist endorses only the potential infinite
VI.3 p.198 Every rational number, unlike every natural number, is divisible by some other number
VII.1 p.215 Limitation of Size is not self-evident, and seems too strong
VII.4 p.226 Second-order set theory just adds a version of Replacement that quantifies over functions
VIII.2 p.248 The infinite is extrapolation from the experience of indefinitely large size
VIII.2 p.256 The theory of infinity must rest on our inability to distinguish between very large sizes