green numbers give full details.
|
back to list of philosophers
|
expand these ideas
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
|
'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal
|
III.4
|
p.53
|
15912
|
Counting results in well-ordering, and well-ordering makes counting possible
|
III.4
|
p.53
|
15913
|
A collection is 'well-ordered' 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 well-ordered, 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
|
Second-order logic presupposes a set of relations already fixed by the first-order 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 set-theory 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
|
Second-order 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
|