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Single Idea 9562
[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
]
Full Idea
In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this).
Gist of Idea
In simple type theory there is a hierarchy of null sets
Source
Charles Chihara (A Structural Account of Mathematics [2004], 07.4)
Book Ref
Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.192
A Reaction
It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong.
The
17 ideas
from 'A Structural Account of Mathematics'
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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
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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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