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Single Idea 10108
[filed under theme 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
]
Full Idea
The problem with reducing arithmetic to ZFC is not that this theory is inconsistent (as far as we know it is not), but rather that is not completely general, and for this reason not logical. For example, it asserts the existence of sets.
Clarification
ZFC is Zermelo-Fraenkel set theory with Choice
Gist of Idea
As a reduction of arithmetic, set theory is not fully general, and so not logical
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.90
A Reaction
Note that ZFC has not been proved consistent.
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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