Editing Empty set (section)

==Questioned existence==
=== Historical issues ===
In the context of sets of real numbers, Cantor used <math>P\equiv O</math> to denote "<math>P</math> contains no single point". This <math>\equiv O</math> notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed <math>O</math> as an existent set on its own, or if Cantor merely used <math>\equiv O</math> as an emptiness predicate. Zermelo accepted <math>O</math> itself as a set, but considered it an "improper set".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/8.pdf The Empty Set, the Singleton, and the Ordered Pair]", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</ref>

=== Axiomatic set theory ===
In [[Zermelo set theory]], the existence of the empty set is assured by the [[axiom of empty set]], and its uniqueness follows from the [[axiom of extensionality]]. However, the axiom of empty set can be shown redundant in at least two ways:
*Standard [[first-order logic]] implies, merely from the [[logical axiom]]s, that {{em|something}} exists, and in the language of set theory, that thing must be a set.  Now the existence of the empty set follows easily from the [[axiom of separation]].
*Even using [[free logic]] (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the [[axiom of infinity]].

=== Philosophical issues ===
While the empty set is a standard and widely accepted mathematical concept, it remains an [[ontological]] curiosity, whose meaning and usefulness are debated by philosophers and logicians.

The empty set is not the same thing as {{em|[[nothing]]}}; rather, it is a set with nothing {{em|inside}} it and a set is always {{em|something}}. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all [[chess opening|opening moves]] in [[chess]] that involve a [[king (chess)|king]]."<ref name="Darling" />

The popular [[syllogism]]
:Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is <math>\varnothing</math>" and the latter to "The set {ham sandwich} is better than the set <math>\varnothing</math>". The first compares elements of sets, while the second compares the sets themselves.<ref name="Darling">{{cite book|title=The Universal Book of Mathematics|author=D. J. Darling|publisher= [[John Wiley and Sons]]|year=2004 |isbn=0-471-27047-4|page=106}}</ref>

[[E. J. Lowe (philosopher)|Jonathan Lowe]] argues that while the empty set
:was undoubtedly an important landmark in the history of mathematics,&nbsp;… we should not assume that its utility in calculation is dependent upon its actually denoting some object.

it is also the case that:
:"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense&mdash;namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a {{em|set}} which has no members. We cannot conjure such an entity into existence by mere stipulation."<ref name="Lowe">{{cite book|title=Locke|author=E. J. Lowe|publisher= [[Routledge]]|year=2005|page=87}}</ref>

[[George Boolos]] argued that much of what has been heretofore obtained by set theory can just as easily be obtained by [[plural quantification]] over individuals, without [[wikt:reification|reifying]] sets as singular entities having other entities as members.<ref>[[George Boolos]] (1984), "To be is to be the value of a variable", ''[[The Journal of Philosophy]]'' 91: 430–49. Reprinted in 1998, ''Logic, Logic and Logic'' ([[Richard Jeffrey]], and Burgess, J., eds.) [[Harvard University Press]], 54–72.</ref>