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In coordinate geometry, points are defined by ordered pairs. However, I have never come across a rigorous definition of what an ordered pairs is, specially given the fact that the same element can occur twice in such a set but not in in sets defined in ZFC.

A.G
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    How about this: an ordered set of length $n$ with elements in a set $X$ is an application $f: {1, 2, \ldots, n} \to X$ – Albert Jun 16 '20 at 12:03
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    A finite ordered set $(a_1, \ldots, a_n)$ (where all $a_i$ belong to $A$) is a function $f : { 1,\ldots, n } \to A$ such that $f(i)=a_i$. – Mauro ALLEGRANZA Jun 16 '20 at 12:04
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    Yes but doesn’t the definition of a function itself require the concept of an ordered set? – A.G Jun 16 '20 at 12:05
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    An ordered pair $(a,b)$ may be defined solely using set theory by ${{a},{a,b}}$ (see this article, for instance). This allows to define functions, hence ordered lists of arbitrary lengths. – Sangchul Lee Jun 16 '20 at 12:06
  • I am not sure of this but in my textbook on calculus the definition of a set supersedes that of a function...maybe its different in logic? – A.G Jun 16 '20 at 12:08
  • @A.G no, it just requires integers to already have been constructed. Note that ${1, \ldots, n}$ is a set, not an ordered set – Albert Jun 16 '20 at 12:09
  • Indeed, a function may be defined using sets. A function $f:A\to B$ between two sets $A$, $B$ is a special type of subset of the Cartesian product $A\times B={(a,b):a\in A, b\in B}$, which only requires the notion of 'ordered pair' and nothing more. – Sangchul Lee Jun 16 '20 at 12:10
  • @sangchul lee, what about the fact that in ZFC the same element cannot occur twice in a set ? – A.G Jun 16 '20 at 12:11
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    When $a = b$, then the "ordered pair" collapses to the singleton ${{a}}$. This should not hinder us from establishing the fact that the ordered pairs so defined satisfies: $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. – Sangchul Lee Jun 16 '20 at 12:14
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    @A.G There is no such fact. It is not forbidden to write ${a}$ as ${a,a}$. Note that ${x\colon P(x)}$ is a class, i.e., a notation that helps us write sentences compactly. Sometimes there exists a set $X$ such that $x\in X$ if and only if $P(x)$, so we can identify $X$ with the class. – Ningxin Jun 16 '20 at 12:17
  • I didn't read every comment. If you know what an ordered pair is, then a (partially) ordered set is a pair $(P, R)$ where $P$ is any set and $R$ is a reflexive, antisymmetric and transitive binary relation of $P$.

    It is common to use the symbol $\leq$ instead of $R$ in this situation (it's just a convention), and it's also common to say $P$ is the ordered set, instead of $(P, \leq)$.

     – Git Gud Jun 16 '20 at 12:36
  • I don't know what you're talking about when you say that the same element can occur twice in an ordered set. – Git Gud Jun 16 '20 at 12:37
  • As an example of the definition above, let $P$ be $\mathbb N$ and $R$ be $\left{(m,n) \colon \exists k\in \mathbb N\left(m+k=n\right)\lor m=n\right}$. Note that $R$ is simply the definition of what it means for a natural number to equal to or smaller than another natural number. – Git Gud Jun 16 '20 at 12:43
  • Yes but only due to convenience of notation..my question was about defining such a set.. – A.G Jun 16 '20 at 13:00
  • Are you saying your question is about ordered pairs? Because it is not, you asked about ordered sets. These are different things. – Git Gud Jun 16 '20 at 13:28
  • FWIW I have been using ordered pairs as a primitive in formal proofs for quite some time now without any problems -- not standard ZFC, sorry. Ordered n-tuples are just seen as a different kind of variable name. It seems to work. See, for example my formal proof of Cantor-Bernstein-Schroeder at http://dcproof.com/CBS.htm See construction of a Cartesian product, lines 29 - 36. See construction of a function as a set of ordered pairs, lines 83 - 86. – Dan Christensen Jun 17 '20 at 02:56

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Points are defined by ordered pairs, not to be confused with an ordered set.

Given two objects $a$ and $b$, the ordered pair $(a, b)$ is rigourously defined as the set $$(a, b) := \big\{ \{a\}, \{a, b\}\}$$

The important facts about ordered pairs are that :

  1. given $a, b$, the ordered pair $(a, b)$ always exist.
  2. one has : $$\boxed{\forall a, b, u, v, \quad (a,b) =(u, v) \Longleftrightarrow a = u \textrm{ and } b = v}$$

An ordered set is a set $X$ together with a binary relation $\operatorname{R}$ on $X$ satisfying :

  1. Reflexivity : $\forall x \in X$, $x\operatorname{R}x$
  2. Antisymmetry : $\forall x, y \in X, \ x\operatorname{R} y \textrm{ and } y \operatorname{R} x \Longrightarrow x = y$
  3. Transitivity : $\forall x, y, z \in X, \ x\operatorname{R} y \textrm{ and } y \operatorname{R} z \Longrightarrow x \operatorname{R} z$

Why "ordered" pair?

Given $a, b$, having the (unordered) pair $\{a, b \}$ doesn't allow to know who is who :
$$\{a, b\} = \{u, v \} \Longleftrightarrow (a =u \textrm{ and } b =v) \textrm{ OR } (a=v \textrm{ and } b = u)$$

In contrast, in an ordered pair$(a, b)$, there is a "first" element (namely $a$) and a "second" element (namely $b$) which are distinguished, whence the name.

Olivier Roche
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  • Thank you...i have now edited it...but what then would you call sets defining points in dimensions greater than 2? – A.G Jun 16 '20 at 14:39
  • @A.G they are called "tuples", or $n$-tuples if you want to emphasize the length $n$ of the tuple. Formally, tuples are defined recursively as follow : the $(n+1)$-tuple $(a_1,\dots, a_{n+1})$ is defined to be the ordered pair $\big(a_1, (a_2,\dots, a_{n+1})\big)$. – Olivier Roche Jun 16 '20 at 15:11