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Single Idea 18151
[filed under theme 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
]
Full Idea
Chihara proposes to replace all sets by reference to the open sentences that define them.
Clarification
'Open sentences' have variables in them
Gist of Idea
Could we replace sets by the open sentences that define them?
Source
report of Charles Chihara (Ontology and the Vicious Circle Principle [1973]) by David Bostock - Philosophy of Mathematics 9.B.4
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.283
A Reaction
This depends on predicativism, because that stipulates the definitions will be available (cos if it ain't definable it ain't there). Chihara went on to define the open sentences in terms of the possibility of uttering them. Cf. propositional functions.
Related Idea
Idea 18136
If we can only think of what we can describe, predicativism may be implied [Bostock]
The
22 ideas
from Charles Chihara
8758
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We could talk of open sentences, instead of sets
[Chihara, by Shapiro]
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10265
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Chihara's system is a variant of type theory, from which he can translate sentences
[Chihara, by Shapiro]
|
8759
|
We can replace type theory with open sentences and a constructibility quantifier
[Chihara, by Shapiro]
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10264
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Introduce a constructibility quantifiers (Cx)Φ - 'it is possible to construct an x such that Φ'
[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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10192
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We can replace existence of sets with possibility of constructing token sentences
[Chihara, by MacBride]
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9547
|
Mathematical entities are causally inert, so the causal theory of reference won't work for them
[Chihara]
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9549
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The set theorist cannot tell us what 'membership' is
[Chihara]
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9550
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We only know relational facts about the empty set, but nothing intrinsic
[Chihara]
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9551
|
What is special about Bill Clinton's unit set, in comparison with all the others?
[Chihara]
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9552
|
Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced
[Chihara]
|
9553
|
Analytic geometry gave space a mathematical structure, which could then have axioms
[Chihara]
|
9559
|
If a successful theory confirms mathematics, presumably a failed theory disconfirms it?
[Chihara]
|
9561
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The mathematics of relations is entirely covered by ordered pairs
[Chihara]
|
9562
|
In simple type theory there is a hierarchy of null sets
[Chihara]
|
9563
|
A pack of wolves doesn't cease when one member dies
[Chihara]
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9566
|
No scientific explanation would collapse if mathematical objects were shown not to exist
[Chihara]
|
9568
|
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
[Chihara]
|
9571
|
ZFU refers to the physical world, when it talks of 'urelements'
[Chihara]
|
9573
|
The null set is a structural position which has no other position in membership relation
[Chihara]
|
9572
|
Realists about sets say there exists a null set in the real world, with no members
[Chihara]
|
9574
|
'Gunk' is an individual possessing no parts that are atoms
[Chihara]
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