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!) The empty set is the unique set with no members:

: <math>\{xemptyset \,y\} \overset{\mathrm{def.}}{=} \{zx \mid z=x \veeneq z=yx\}</math>

The empty set is the unique set with no members. In NFU, there are also urelements with no members.

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

: <math>\displaystyle\{x \cup y} \overset{\mathrm{def.}}{=} \{zy \mid zy \in= x \vee z \in y\}</math>

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!)

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>\emptyset \{x,y\} \overset{\mathrm{def.}}{=} \{xz \mid z=x \neqvee xz=y\}</math>

The union of two sets is defined in the usual way.:

: <math>x \{x_1,\ldots,x_n,x_{n+1}\}cup y \overset{\mathrm{def.}}{=} \{x_1,z \ldots,x_nmid z \}in x \cupvee z \{x_{n+1}in y\}</math>

The empty set is the unique set with no members. In NFU, there are also urelements with no members.

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>\displaystyle{x_1,\ldots,x_n,x_{xn+1}\} \overset{\mathrm{def.}}{=} \{yx_1,\ldots,x_n\} \midcup y = x\{x_{n+1}\}</math>

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

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

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

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

The union of two sets is defined in the usual way.

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

This is a recursive definition of unordered ''n''-tuples for any concrete ''n'' (finite sets given as lists of their elements).

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 of Pairing, the existence of the empty set follows by Separation from the existence of any set, and the boolean union of two sets exists by the axioms of Pairing and Union (<math>x \cup y = \bigcup\{x,y\}</math>).