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Single Idea 8745
[filed under theme 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
]
Full Idea
The symbols for classes are mere conveniences, not representing objects called 'classes'. Classes are in fact logical fictions; they cannot be regarded as part of the ultimate furniture of the world.
Gist of Idea
Classes are logical fictions, and are not part of the ultimate furniture of the world
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], Ch.18), quoted by Stewart Shapiro - Thinking About Mathematics 5.2
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.123
A Reaction
I agree. For 'logical fictions' read 'abstractions'. To equate abstractions with fictions is to underline the fact that they are a human creation. They are either that or platonic objects - there is no middle way.
The
23 ideas
with the same theme
[objections to the whole idea of set theory]:
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8745
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Classes are logical fictions, and are not part of the ultimate furniture of the world
[Russell]
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6436
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I gradually replaced classes with properties, and they ended as a symbolic convenience
[Russell]
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10044
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Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element
[Russell/Whitehead, by Shapiro]
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18208
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We regard classes as mere symbolic or linguistic conveniences
[Russell/Whitehead]
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11064
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Classes can be reduced to propositional functions
[Russell, by Hanna]
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7548
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Classes, grouped by a convenient property, are logical constructions
[Russell]
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13536
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Skolem did not believe in the existence of uncountable sets
[Skolem]
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10304
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Very few things in set theory remain valid in intuitionist mathematics
[Bernays]
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3355
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Von Neumann wanted mathematical functions to replace sets
[Neumann, by Benardete,JA]
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9920
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Two objects can apparently make up quite distinct arrangements in sets
[Goodman, by Burgess/Rosen]
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3336
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Two things can never entail three things
[Quine, by Benardete,JA]
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10699
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Does a bowl of Cheerios contain all its sets and subsets?
[Boolos]
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8972
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What in the real world could ground the distinction between the sets {A,{A,B}} and {B,{A,B}}?
[Inwagen]
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9570
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In Field's Platonist view, set theory is false because it asserts existence for non-existent things
[Field,H, by Chihara]
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6548
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Physicalism requires the naturalisation or rejection of set theory
[Lycan]
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13481
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Maybe sets should be rethought in terms of the even more basic categories
[Hart,WD]
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7035
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God does not create the world, and then add the classes
[Heil]
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10207
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Anti-realists reject set theory
[Shapiro]
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8758
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We could talk of open sentences, instead of sets
[Chihara, by Shapiro]
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18151
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Could we replace sets by the open sentences that define them?
[Chihara, by Bostock]
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9563
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A pack of wolves doesn't cease when one member dies
[Chihara]
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10108
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As a reduction of arithmetic, set theory is not fully general, and so not logical
[George/Velleman]
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10687
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Maybe we reduce sets to ordinals, rather than the other way round
[Hossack]
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