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Single Idea 17752
[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
]
Full Idea
The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are.
Gist of Idea
The empty set is useful for defining sets by properties, when the members are not yet known
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 1.1)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.45
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]
|
15502
|
There are four main reasons for asserting that there is an empty set
[Lewis]
|
15503
|
We needn't accept this speck of nothingness, this black hole in the fabric of Reality!
[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]
|
9572
|
Realists about sets say there exists a null set in the real world, with no members
[Chihara]
|
9573
|
The null set is a structural position which has no other position in membership relation
[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]
|
14239
|
The empty set is usually derived from Separation, but it also seems to need Infinity
[Oliver/Smiley]
|
14242
|
Maybe we can treat the empty set symbol as just meaning an empty term
[Oliver/Smiley]
|
14240
|
The empty set is something, not nothing!
[Oliver/Smiley]
|
14241
|
We don't need the empty set to express non-existence, as there are other ways to do that
[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]
|