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Single Idea 15498
[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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Full Idea
There is no such class as the null class. I don't mind calling some memberless thing - some individual - the null 'set'. But that doesn't make it a memberless class.
Gist of Idea
We can accept the null set, but there is no null class of anything
Source
David Lewis (Parts of Classes [1991], 1.2)
Book Ref
Lewis,David: 'Parts of Classes' [Blackwell 1991], p.4
A Reaction
The point is that set theory is a formal system which can do what it likes, but classes are classes 'of' things. Everyone assumes that sets are classes, reserving 'proper classes' for the tricky cases up at the far end.
The
35 ideas
from 'Parts of Classes'
18395
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Sets are mereological sums of the singletons of their members
[Lewis, by Armstrong]
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10566
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Lewis prefers giving up singletons to giving up sums
[Lewis, by Fine,K]
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14244
|
Lewis only uses fusions to create unities, but fusions notoriously flatten our distinctions
[Oliver/Smiley on Lewis]
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10191
|
Set theory reduces to a mereological theory with singletons as the only atoms
[Lewis, by MacBride]
|
15497
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We can replace the membership relation with the member-singleton relation (plus mereology)
[Lewis]
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15496
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We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton
[Lewis]
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15500
|
Classes divide into subclasses in many ways, but into members in only one way
[Lewis]
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15499
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A subclass of a subclass is itself a subclass; a member of a member is not in general a member
[Lewis]
|
15498
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We can accept the null set, but there is no null class of anything
[Lewis]
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15501
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We have no idea of a third sort of thing, that isn't an individual, a class, or their mixture
[Lewis]
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15503
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We needn't accept this speck of nothingness, this black hole in the fabric of Reality!
[Lewis]
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15502
|
There are four main reasons for asserting that there is an empty set
[Lewis]
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15504
|
Atomless gunk is an individual whose parts all have further proper parts
[Lewis]
|
15506
|
If we don't understand the singleton, then we don't understand classes
[Lewis]
|
15507
|
Set theory has some unofficial axioms, generalisations about how to understand it
[Lewis]
|
15508
|
If singletons are where their members are, then so are all sets
[Lewis]
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15509
|
Some say qualities are parts of things - as repeatable universals, or as particulars
[Lewis]
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15511
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If singleton membership is external, why is an object a member of one rather than another?
[Lewis]
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15512
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In mereology no two things consist of the same atoms
[Lewis]
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15513
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Maybe singletons have a structure, of a thing and a lasso?
[Lewis]
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15514
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A huge part of Reality is only accepted as existing if you have accepted set theory
[Lewis]
|
15515
|
To be a structuralist, you quantify over relations
[Lewis]
|
15516
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A property is any class of possibilia
[Lewis]
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15517
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Giving up classes means giving up successful mathematics because of dubious philosophy
[Lewis]
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15518
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I like plural quantification, but am not convinced of its connection with second-order logic
[Lewis]
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15520
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Existence doesn't come in degrees; once asserted, it can't then be qualified
[Lewis]
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15519
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Trout-turkeys exist, despite lacking cohesion, natural joints and united causal power
[Lewis]
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15521
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Given cats, a fusion of cats adds nothing further to reality
[Lewis]
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15522
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The one has different truths from the many; it is one rather than many, one rather than six
[Lewis]
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14748
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The many are many and the one is one, so they can't be identical
[Lewis]
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15523
|
Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it
[Lewis]
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15524
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Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory
[Lewis]
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15525
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Plural quantification lacks a complete axiom system
[Lewis]
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10660
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A commitment to cat-fusions is not a further commitment; it is them and they are it
[Lewis]
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6129
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Lewis affirms 'composition as identity' - that an object is no more than its parts
[Lewis, by Merricks]
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