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Single Idea 13199

[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set ]

Full Idea

It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed.

Gist of Idea

The empty set may look pointless, but many sets can be constructed from it

Source

Herbert B. Enderton (Elements of Set Theory [1977], 1:02)

Book Ref

Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.2

A Reaction

This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction.

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]
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]
14242 Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley]
14239 The empty set is usually derived from Separation, but it also seems to need Infinity [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]
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]