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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

18250

Cauchy gave a necessary condition for the convergence of a sequence

I

p.4

15898

The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules

I

p.5

15899

Replacement was immediately accepted, despite having very few implications

I

p.5

15900

The iterative conception of set wasn't suggested until 1947

II.6

p.38

15904

The two sides of the Cut are, roughly, the bounding commensurable ratios

III.2

p.47

15907

Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity

III.3

p.50

15909

'Aleph0' is cardinality of the naturals, 'aleph1' the next cardinal, 'alephω' the ωth cardinal

III.4

p.53

15912

Counting results in wellordering, and wellordering makes counting possible

III.4

p.53

15913

A collection is 'wellordered' if there is a least element, and all of its successors can be identified

III.4

p.53

15914

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.54

15915

Ordinals are basic to Cantor's transfinite, to count the sets

III.5

p.61

15917

Paradox: the class of all ordinals is wellordered, so must have an ordinal as type  giving a bigger ordinal

III.5

p.62

15918

Paradox: there is no largest cardinal, but the class of everything seems to be the largest

IV.1

p.63

15919

The 'logical' notion of class has some kind of definition or rule to characterise the class

IV.2

p.78

15921

Collections of things can't be too big, but collections by a rule seem unlimited in size

IV.2

p.78

15920

Pure collections of things obey Choice, but collections defined by a rule may not

IV.2

p.92

15922

For the real numbers to form a set, we need the Continuum Hypothesis to be true

V.3

p.123

15926

Secondorder logic presupposes a set of relations already fixed by the firstorder domain

V.3

p.133

15929

Set theory will found all of mathematics  except for the notion of proof

V.3 n33

p.132

15928

Intuitionism rejects settheory to found mathematics

V.4

p.135

15930

Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets

V.5

p.148

15931

The iterative conception needs the Axiom of Infinity, to show how far we can iterate

V.5

p.149

15932

The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs

V.5

p.150

15933

Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement

VI.1

p.155

15934

Mathematical proof by contradiction needs the law of excluded middle

VI.1

p.157

15935

Modern mathematics works up to isomorphism, and doesn't care what things 'really are'

VI.1

p.160

15936

The Power Set is just the collection of functions from one collection to another

VI.2

p.164

15937

Those who reject infinite collections also want to reject the Axiom of Choice

VI.2

p.176

15940

The intuitionist endorses only the potential infinite

VI.3

p.198

15942

Every rational number, unlike every natural number, is divisible by some other number

VII.4

p.226

15945

Secondorder set theory just adds a version of Replacement that quantifies over functions

VIII.2

p.248

15947

The infinite is extrapolation from the experience of indefinitely large size

VIII.2

p.256

15949

The theory of infinity must rest on our inability to distinguish between very large sizes
