Editing Empty set (section)

=== 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>