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Single Idea 15929
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
]
Full Idea
Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof.
Gist of Idea
Set theory will found all of mathematics - except for the notion of proof
Source
Shaughan Lavine (Understanding the Infinite [1994], V.3)
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.133
The
33 ideas
from Shaughan Lavine
18250
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Cauchy gave a necessary condition for the convergence of a sequence
[Lavine]
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15899
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Replacement was immediately accepted, despite having very few implications
[Lavine]
|
15898
|
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules
[Lavine]
|
15900
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The iterative conception of set wasn't suggested until 1947
[Lavine]
|
15904
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The two sides of the Cut are, roughly, the bounding commensurable ratios
[Lavine]
|
15907
|
Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity
[Lavine]
|
15909
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'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal
[Lavine]
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15912
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Counting results in well-ordering, and well-ordering makes counting possible
[Lavine]
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15915
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Ordinals are basic to Cantor's transfinite, to count the sets
[Lavine]
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15914
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An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one
[Lavine]
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15913
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A collection is 'well-ordered' if there is a least element, and all of its successors can be identified
[Lavine]
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15918
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Paradox: there is no largest cardinal, but the class of everything seems to be the largest
[Lavine]
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15917
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Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal
[Lavine]
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15919
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The 'logical' notion of class has some kind of definition or rule to characterise the class
[Lavine]
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15921
|
Collections of things can't be too big, but collections by a rule seem unlimited in size
[Lavine]
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15920
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Pure collections of things obey Choice, but collections defined by a rule may not
[Lavine]
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15922
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For the real numbers to form a set, we need the Continuum Hypothesis to be true
[Lavine]
|
15929
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Set theory will found all of mathematics - except for the notion of proof
[Lavine]
|
15926
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Second-order logic presupposes a set of relations already fixed by the first-order domain
[Lavine]
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15928
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Intuitionism rejects set-theory to found mathematics
[Lavine]
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15930
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Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets
[Lavine]
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15931
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The iterative conception needs the Axiom of Infinity, to show how far we can iterate
[Lavine]
|
15932
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The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs
[Lavine]
|
15933
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Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement
[Lavine]
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15934
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Mathematical proof by contradiction needs the law of excluded middle
[Lavine]
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15936
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The Power Set is just the collection of functions from one collection to another
[Lavine]
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15935
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Modern mathematics works up to isomorphism, and doesn't care what things 'really are'
[Lavine]
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15940
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The intuitionist endorses only the potential infinite
[Lavine]
|
15937
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Those who reject infinite collections also want to reject the Axiom of Choice
[Lavine]
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15942
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Every rational number, unlike every natural number, is divisible by some other number
[Lavine]
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15945
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Second-order set theory just adds a version of Replacement that quantifies over functions
[Lavine]
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15947
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The infinite is extrapolation from the experience of indefinitely large size
[Lavine]
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15949
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The theory of infinity must rest on our inability to distinguish between very large sizes
[Lavine]
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