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Single Idea 9933
[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
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Full Idea
The paradoxes only seem to arise in connection with Frege's logical notion of extension or class, not Cantor's mathematical notion of set. Cantor never assumed that every condition determines a set.
Gist of Idea
The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set
Source
JP Burgess / G Rosen (A Subject with No Object [1997], III.C.1.b)
Book Ref
Burgess,J/Rosen,G: 'A Subject with No Object' [OUP 1997], p.224
A Reaction
This makes the whole issue a parochial episode in the history of philosophy, not a central question. Cantor favoured some sort of abstractionism (see Kit Fine on the subject).
The
15 ideas
from 'A Subject with No Object'
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9918
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Abstract/concrete is a distinction of kind, not degree
[Burgess/Rosen]
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9919
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The old debate classified representations as abstract, not entities
[Burgess/Rosen]
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9921
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'True' is only occasionally useful, as in 'everything Fermat believed was true'
[Burgess/Rosen]
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9922
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If space is really just a force-field, then it is a physical entity
[Burgess/Rosen]
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9923
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We should talk about possible existence, rather than actual existence, of numbers
[Burgess/Rosen]
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9924
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Modal logic gives an account of metalogical possibility, not metaphysical possibility
[Burgess/Rosen]
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9925
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Structuralism and nominalism are normally rivals, but might work together
[Burgess/Rosen]
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9926
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A relation is either a set of sets of sets, or a set of sets
[Burgess/Rosen]
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9928
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Mereology implies that acceptance of entities entails acceptance of conglomerates
[Burgess/Rosen]
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9927
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Mathematics has ascended to higher and higher levels of abstraction
[Burgess/Rosen]
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9929
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Much of what science says about concrete entities is 'abstraction-laden'
[Burgess/Rosen]
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9930
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Abstraction is on a scale, of sets, to attributes, to type-formulas, to token-formulas
[Burgess/Rosen]
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9933
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The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set
[Burgess/Rosen]
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9932
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The paradoxes no longer seem crucial in critiques of set theory
[Burgess/Rosen]
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9934
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Number words became nouns around the time of Plato
[Burgess/Rosen]
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