Implementation of mathematics in set theory: Difference between revisions

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== Empty set, singleton, unordered pairs and tuples ==

These constructions appear first because they are the simplest constructions in set theory, not because they are the first constructions that come to mind in mathematics (though the notion of finite set is certainly fundamental!). Even though NFU also allows the construction of set [[Urelement|ur-elements]] yet to become members of a set, the [[empty set]] is the unique ''set'' with no members:

:<math>\left.\varnothing\right. \, \overset{\mathrm{def.}}{=} \left\{x : x \neq x\right\}</math>

Even though NFU also allows the construction of set [[Urelement|ur-elements]] yet to become members of a set, the [[empty set]] is the unique ''set'' with no members:

:<math>\left.\varnothing\right. \overset{\mathrm{def.}}{=} \left\{x : x \neq x\right\}</math>

For each object <math>x</math>, there is a set <math>\{x\}</math> with <math>x</math> as its only element:

:<math>\left\{x\right\} \overset{\mathrm{def.}}{=} \left\{y : y = x\right\}</math>

For objects <math>x</math> and <math>y</math>, there is a set <math>\{x,y\}</math> containing <math>x</math> and <math>y</math> as its only elements:

:<math>\left\{x,y\right\} \overset{\mathrm{def.}}{=} \left\{z : z=x \vee z = y\right\}</math>

The [[Union (set theory)|union]] of two sets is defined in the usual way:

:<math>\left.x \cup y\right. \, \overset{\mathrm{def.}}{=} \left\{z : z \in x \vee z \in y\right\}</math>

This is a recursive definition of unordered <math>n</math>-tuples for any concrete <math>n</math> (finite sets given as lists of their elements:)

:<math>\left\{x_1, \ldots, x_n, x_{n+1}\right\} \overset{\mathrm{def.}}{=} \left\{x_1, \ldots, x_n\right\} \cup \left\{x_{n+1}\right\}</math>

In [[New Foundations|NFU]], all the set definitions given work by stratified comprehension; in [[ZFC]], the existence of the unordered pair is given by the axiom[[Axiom of pairing|Axiom of Pairing]], the existence of the empty set follows by [[Axiom schema of separation|Separation]] from the existence of any set, and the booleanbinary union of two sets exists by the axioms of Pairing and [[Axiom of union|Union]] (<math>x \cup y = \bigcup\{x,y\}</math>).

== Ordered pair ==