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Single Idea 14113
[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
]
Full Idea
The null class is a fiction.
Gist of Idea
The null class is a fiction
Source
Bertrand Russell (The Principles of Mathematics [1903], §079)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.81
A Reaction
This does not commit him to regarding all classes as fictions - though he seems to have eventually come to believe that. The null class seems to have a role something like 'Once upon a time...' in story-telling. You can then tell truth or fiction.
The
30 ideas
with the same theme
[status of a set having no members]:
14238
|
A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class
[Frege]
|
9157
|
The null set is only defensible if it is the extension of an empty concept
[Frege, by Burge]
|
9835
|
It is because a concept can be empty that there is such a thing as the empty class
[Frege, by Dummett]
|
16895
|
The null set is indefensible, because it collects nothing
[Frege, by Burge]
|
21548
|
The null class is the class with all the non-existents as its members
[MacColl, by Lackey]
|
14113
|
The null class is a fiction
[Russell]
|
18393
|
For 'there is a class with no members' we don't need the null set as truthmaker
[Armstrong]
|
13200
|
Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ
[Enderton]
|
13199
|
The empty set may look pointless, but many sets can be constructed from it
[Enderton]
|
10809
|
We can accept the null set, but not a null class, a class lacking members
[Lewis]
|
10811
|
The null set plays the role of last resort, for class abstracts and for existence
[Lewis]
|
10812
|
The null set is not a little speck of sheer nothingness, a black hole in Reality
[Lewis]
|
15498
|
We can accept the null set, but there is no null class of anything
[Lewis]
|
15503
|
We needn't accept this speck of nothingness, this black hole in the fabric of Reality!
[Lewis]
|
15502
|
There are four main reasons for asserting that there is an empty set
[Lewis]
|
13442
|
Without the empty set we could not form a∩b without checking that a and b meet
[Hart,WD]
|
9986
|
The null set was doubted, because numbering seemed to require 'units'
[Tait]
|
9550
|
We only know relational facts about the empty set, but nothing intrinsic
[Chihara]
|
9562
|
In simple type theory there is a hierarchy of null sets
[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]
|
8322
|
I don't believe in the empty set, because (lacking members) it lacks identity-conditions
[Lowe]
|
10713
|
Usually the only reason given for accepting the empty set is convenience
[Potter]
|
14240
|
The empty set is something, not nothing!
[Oliver/Smiley]
|
14239
|
The empty set is usually derived from Separation, but it also seems to need Infinity
[Oliver/Smiley]
|
14241
|
We don't need the empty set to express non-existence, as there are other ways to do that
[Oliver/Smiley]
|
14242
|
Maybe we can treat the empty set symbol as just meaning an empty term
[Oliver/Smiley]
|
8677
|
Set theory makes a minimum ontological claim, that the empty set exists
[Friend]
|
17752
|
The empty set is useful for defining sets by properties, when the members are not yet known
[Walicki]
|
17753
|
The empty set avoids having to take special precautions in case members vanish
[Walicki]
|