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Single Idea 9552
[filed under theme 5. Theory of Logic / K. Features of Logics / 2. Consistency
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Full Idea
In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it.
Gist of Idea
Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced
Source
Charles Chihara (A Structural Account of Mathematics [2004], 02.1)
Book Ref
Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.33
A Reaction
The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work.
The
17 ideas
from 'A Structural Account of Mathematics'
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
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Mathematical entities are causally inert, so the causal theory of reference won't work for them
[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
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What is special about Bill Clinton's unit set, in comparison with all the others?
[Chihara]
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9549
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The set theorist cannot tell us what 'membership' is
[Chihara]
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9552
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Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced
[Chihara]
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9553
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Analytic geometry gave space a mathematical structure, which could then have axioms
[Chihara]
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9559
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If a successful theory confirms mathematics, presumably a failed theory disconfirms it?
[Chihara]
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9561
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The mathematics of relations is entirely covered by ordered pairs
[Chihara]
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9562
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In simple type theory there is a hierarchy of null sets
[Chihara]
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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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9566
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No scientific explanation would collapse if mathematical objects were shown not to exist
[Chihara]
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9568
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I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
[Chihara]
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9571
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ZFU refers to the physical world, when it talks of 'urelements'
[Chihara]
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9573
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The null set is a structural position which has no other position in membership relation
[Chihara]
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9572
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Realists about sets say there exists a null set in the real world, with no members
[Chihara]
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9574
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'Gunk' is an individual possessing no parts that are atoms
[Chihara]
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