Mathematics is based in fundamental agreements, the most fundamental one being the existence of sets and elements, with the property that sets contain elements. It must be given that for all elements \(x\) and for a set \(X\), either \(x\in X\) or \(x\notin X\). Let \(X=\{a,b\}\); then \(a\in X\) and \(b\in X\). We look to expand our objects in order to represent order of elements.
Definition (Kuratowski, 1921): An ordered pair is the set \[ (x,y)=\{\,x,\{x,y\}\,\}. \] It follows that \((x,y)\neq (y,x)\).
Proposition: \(((x,y)=(a,b))\iff (x=a\text{ and }y=b)\).
\(\Leftarrow\) Assume \(x=a\) and \(y=b\). Substituting into the left‑hand side
gives \((x,y)=(a,b)\), i.e. \((a,b)=(a,b)\), which is true.
\(\Rightarrow\) Assume \((x,y)=(a,b)\). By definition,
\[
\{\,a,\{a,b\}\,\}=\{\,x,\{x,y\}\,\}.
\]
Comparing the two–element sets we obtain the two cases below.
Case 1. \(a=b\). Then \((a,b)=\{a,\{a\}\}=\{a\}\). Hence \(\{x,\{x,y\}\}=\{a\}\) which forces
\(x=y=a\) and therefore \(a=x\) and \(b=y\).
Case 2. \(a\neq b\). Then the unordered pairs \(\{a,\{a,b\}\}\) and
\(\{x,\{x,y\}\}\) each contain two distinct elements. Equality of the two sets forces
\(a=x\) and \(\{a,b\}=\{x,y\}\), whence \(b=y\). ∎
Definition: Let \(X,Y\) be sets. The Cartesian product of \(X\) and \(Y\) is \[ X\times Y=\{\, (x,y)\mid x\in X\text{ and }y\in Y\,\}. \]
Definition: A relation \(R\) on a set \(X\) is a subset \(R\subseteq X\times X\).
For example, with \(X=\{a,b,c\}\) we have \[ X\times X=\{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)\}. \] Two particular relations on \(X\) are \[ R_{1}=\{(a,a),(b,b),(c,c)\},\qquad R_{2}=\{(a,b),(b,c),(c,c)\}. \]
Definition: Let \(X\neq\varnothing\). A **partial order** on \(X\) is a relation \(R\subseteq X\times X\) satisfying \[ \begin{aligned} &\text{Reflexivity: } &&\forall x\in X\;(x,x)\in R,\\[2pt] &\text{Antisymmetry: } &&\forall x,y\in X\;[(x,y)\in R\land(y,x)\in R\;\Rightarrow\;x=y],\\[2pt] &\text{Transitivity: } &&\forall x,y,z\in X\;[(x,y)\in R\land(y,z)\in R\;\Rightarrow\;(x,z)\in R]. \end{aligned} \]
For the set \(X=\{a,b,c\}\) the relation \[ R=\{(a,a),(a,b),(a,c),(b,b),(b,c),(c,c)\} \] is a partial order. It is reflexive, it is antisymmetric because none of the pairs \((x,y)\) and \((y,x)\) occur simultaneously with \(x\neq y\), and it is transitive (e.g. \((a,b),(b,c)\in R\Rightarrow (a,c)\in R\)). We write \(x\preccurlyeq y\) for \((x,y)\in R\). The familiar “\(\le\)” on \(\mathbb N\) is a partial order.
Definition: A relation \(R\subseteq X\times X\) is an **equivalence relation** when it is reflexive, symmetric, and transitive.
Number systems can be created through multiple definitions; we will use the Peano axioms together with an explicit set‑theoretic construction. Let
\[ \begin{aligned} \mathbb N_{0}&=\varnothing,\\ \mathbb N_{1}&=\{\varnothing\},\\ \mathbb N_{2}&=\{\varnothing,\{\varnothing\}\},\\ \mathbb N_{3}&=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\},\dots\}, \end{aligned} \qquad \mathbb N_{n+1}= \mathbb N_{n}\cup\{\mathbb N_{n}\}. \] The object \(\mathbb N=\bigcup_{n\in\mathbb N} \mathbb N_{n}\) is the set of natural numbers.
Definition (Peano Axioms): Let \(\mathbb N\) be a set with a distinguished element \(0\in\mathbb N\) and a function \(s:\mathbb N\to\mathbb N\) (the **successor**). The triple \((\mathbb N,s,0)\) satisfies the Peano axioms when \[ \begin{aligned} &(1)\qquad &\forall n\in\mathbb N\;[\,s(n)\neq 0\,];\\[2pt] &(2)\qquad &\forall m,n\in\mathbb N\;[\,s(m)=s(n)\;\Rightarrow\;m=n\,];\\[2pt] &(3)\qquad &\text{If }K\subseteq\mathbb N,\;0\in K\text{ and }(\forall n\in K\;s(n)\in K),\text{ then }K=\mathbb N. \end{aligned} \]
Property (3) is the induction principle: to prove a statement \(P(n)\) for all \(n\in\mathbb N\) we show that \(P(0)\) holds and that \(P(k)\Rightarrow P(s(k))\) for an arbitrary \(k\). The set of all \(n\) for which \(P(n)\) holds is inductive, hence equal to \(\mathbb N\).
Let \(K=\{\,n\in\mathbb N\mid P(n)\,\}\). Showing \(0\in K\) (base step) and \(\forall m\in K\;(s(m)\in K)\) (inductive step) yields \(K\) inductive, so by (3) we have \(K=\mathbb N\); therefore \(P(n)\) holds for every natural number.
Definition: A **binary operation** on a set \(X\) is a map \(\theta:X\times X\to X\).
Definition: Let \(\mathbb N\) be the set of natural numbers. Define an equivalence relation \(\sim\) on \(\mathbb N\times\mathbb N\) by \[ (a,b)\sim(c,d)\;\Longleftrightarrow\;a+d=b+c . \] The **integers** are the equivalence classes: \[ \mathbb Z:= (\mathbb N\times\mathbb N)/\!\sim . \] The class of \((a,b)\) is denoted \([a,b]\) and is interpreted as the “difference’’ \(a-b\).
Reflexivity. For any \((a,b)\) we have \(a+b=b+a\); hence
\((a,b)\sim(a,b)\).
Symmetry. If \((a,b)\sim(c,d)\) then \(a+d=b+c\), which is equivalent to
\(c+b=d+a\); thus \((c,d)\sim(a,b)\).
Transitivity. Suppose \((a,b)\sim(c,d)\) and \((c,d)\sim(e,f)\). Then
\(a+d=b+c\) and \(c+f=d+e\). Adding the two equalities gives
\(a+d+c+f = b+c+d+e\), i.e. \(a+f = b+e\); hence \((a,b)\sim(e,f)\). ∎
Definition (Addition on \(\mathbb Z\)): For \([a,b],[c,d]\in\mathbb Z\) set \[ [a,b]\;+\;[c,d]\;:=\;[a+c,\;b+d]. \]
Assume \([a,b]=[a',b']\) and \([c,d]=[c',d']\); thus \(a+b'=b+a'\) and \(c+d'=d+c'\). Then \[ (a+c)+(b'+d') = (a+b')+(c+d') = (b+a')+(d+c') = (b+d)+(a'+c'), \] which shows \([a+c,b+d]=[a'+c',b'+d']\). Hence the definition does not depend on the representatives. ∎
Definition (Multiplication on \(\mathbb Z\)): For \([a,b],[c,d]\in\mathbb Z\) set \[ [a,b]\;\cdot\;[c,d]\;:=\;[ac+bd,\;ad+bc]. \]
Suppose \([a,b]=[a',b']\) and \([c,d]=[c',d']\). Then \(a+b'=b+a'\) and \(c+d'=d+c'\). A short computation using distributivity gives \[ (ac+bd)+(a'c'+b'd') = (a+b')(c+d') = (b+a')(d+c') = (ad+bc)+(a'd'+b'c'). \] Thus \([ac+bd,ad+bc]=[a'c'+b'd',a'd'+b'c']\), and the product is independent of the representatives. ∎
Definition (Order on \(\mathbb Z\)): For \([a,b],[c,d]\in\mathbb Z\) write \[ [a,b]\;\le\;[c,d]\quad\Longleftrightarrow\quad a+d\le_{\mathbb N} c+b, \] where \(\le_{\mathbb N}\) is the usual order on natural numbers.
If \([a,b]=[a',b']\) and \([c,d]=[c',d']\) then \(a+b'=b+a'\) and \(c+d'=d+c'\). Adding the first equality to \(c+d\) and the second to \(a+b\) yields \[ a+d+b'+c = a+b'+c+d = b+a'+c+d = b+c+d'+a' . \] Hence \(a+d\le c+b\) iff \(a'+d'\le c'+b'\). Thus the relation does not depend on the chosen representatives. ∎
Definition: Let \(\mathbb Z\) be the set of integers. Define an equivalence relation \(\approx\) on \(\mathbb Z\times(\mathbb Z\setminus\{0\})\) by \[ (p,q)\approx(r,s)\;\Longleftrightarrow\;ps=qr . \] The **rationals** are the equivalence classes \[ \mathbb Q:=\bigl(\mathbb Z\times(\mathbb Z\setminus\{0\})\bigr)\big/\!\approx . \] The class of \((p,q)\) is denoted \([p,q]\) and is interpreted as the fraction \(\frac{p}{q}\).
Reflexivity: \(p q = q p\) gives \((p,q)\approx(p,q)\).
Symmetry: from \(ps=qr\) we obtain \(qr=ps\), so \((r,s)\approx(p,q)\).
Transitivity: if \((p,q)\approx(r,s)\) and \((r,s)\approx(u,v)\) then
\(ps=qr\) and \(rv=su\). Multiplying the first equality by \(v\) and the second by \(q\) gives
\(psv=qr v\) and \(q rv = q s u\). Since \(qr v = q r v\), we obtain \(psv = q s u\),
i.e. \(p v = q u\). Hence \((p,q)\approx(u,v)\). ∎
Definition (Addition on \(\mathbb Q\)): For \([p,q],[r,s]\in\mathbb Q\) set \[ [p,q]\;+\;[r,s]\;:=\;[ps+rq,\;qs]. \]
Suppose \((p,q)\approx(p',q')\) and \((r,s)\approx(r',s')\); thus \(pq'=p'q\) and \(rs'=r's\). Compute \[ (ps+rq)s'q' = p s s' q' + r q s' q' = p' q s s' + r' q' s s \] which equals \((p's'+r'q')q s\). Hence the two representatives give the same class, showing the operation is independent of the chosen representatives. ∎
Definition (Multiplication on \(\mathbb Q\)): For \([p,q],[r,s]\in\mathbb Q\) set \[ [p,q]\;\cdot\;[r,s]\;:=\;[pr,\;qs]. \]
Using the same hypotheses as above, \[ (pr)q's' = p q' r s' = p' q r' s = (p'q') (r's') \] and thus the product does not depend on the representatives. ∎
Definition (Order on \(\mathbb Q\)): For \([p,q],[r,s]\in\mathbb Q\) with \(q>0,\;s>0\) define \[ [p,q]\;\le\;[r,s]\quad\Longleftrightarrow\quad ps\le qr, \] where the inequality on the right‑hand side is the usual order on \(\mathbb Z\).
If \((p,q)\approx(p',q')\) then \(pq'=p'q\); similarly for \((r,s)\approx(r',s')\). Multiplying the equality \(ps\le qr\) by the positive integer \(q's'\) yields \(p s q' s' \le q r q' s'\). Substituting using the equivalence equalities gives \(p' s' q s \le q' r' s q'\); which is precisely \(p's' \le q'r'\). Hence the order does not depend on the representatives. ∎
Definition (Sequence of rationals): A **sequence** of rational numbers is a function \(\alpha:\mathbb N\to\mathbb Q\). We write \(\alpha=(a_{n})_{n\in\mathbb N}\) with \(a_{n}=\alpha(n)\).
Definition (Absolute value on \(\mathbb Q\)): For a rational \(q=\frac{p}{r}\) with \(p\in\mathbb Z,\;r\in\mathbb N\setminus\{0\}\) define \[ |q|:=\frac{|p|}{r}, \] where \(|p|\) is the usual absolute value on integers.
Definition (Cauchy sequence): A sequence \((a_{n})\) of rationals is **Cauchy** if \[ \forall\varepsilon>0\; \exists N\in\mathbb N\; \forall m,n\ge N\; |a_{m}-a_{n}|<\varepsilon . \] Here \(\varepsilon\) ranges over the positive rationals.
Definition (Null sequence): A Cauchy sequence \((a_{n})\) is a **null sequence** (or **converges to \(0\)**) when \[ \forall\varepsilon>0\; \exists N\in\mathbb N\; \forall n\ge N\; |a_{n}|<\varepsilon . \]
Definition (Equivalence of Cauchy sequences): For Cauchy sequences \(\alpha=(a_{n})\) and \(\beta=(b_{n})\) define \[ \alpha\sim\beta \quad\Longleftrightarrow\quad (a_{n}-b_{n})_{n\in\mathbb N} \text{ is a null sequence.} \]
Reflexivity. \(a_{n}-a_{n}=0\) for all \(n\); the zero sequence is null,
so \(\alpha\sim\alpha\).
Symmetry. If \(\alpha\sim\beta\) then \((a_{n}-b_{n})\) is null.
Multiplying by \(-1\) shows \((b_{n}-a_{n})\) is null; hence \(\beta\sim\alpha\).
Transitivity. Suppose \(\alpha\sim\beta\) and \(\beta\sim\gamma\).
Then \((a_{n}-b_{n})\) and \((b_{n}-c_{n})\) are null. Given \(\varepsilon>0\) choose
\(N_{1},N_{2}\) such that for all \(n\ge N_{1}\) we have \(|a_{n}-b_{n}|<\varepsilon/2\)
and for all \(n\ge N_{2}\) we have \(|b_{n}-c_{n}|<\varepsilon/2\). For
\(n\ge\max(N_{1},N_{2})\) the triangle inequality yields
\(|a_{n}-c_{n}|\le|a_{n}-b_{n}|+|b_{n}-c_{n}|<\varepsilon\). Hence
\((a_{n}-c_{n})\) is null and \(\alpha\sim\gamma\). ∎
Definition (Real numbers): Let \(\mathcal C\) be the set of all Cauchy sequences of rational numbers. The **real numbers** are the set of equivalence classes \[ \mathbb R:=\mathcal C\big/\!\sim . \] An element of \(\mathbb R\) is denoted by \([\alpha]\) where \(\alpha\) is a Cauchy sequence of rationals. We call \([\alpha]\) the **real number represented by \(\alpha\)**.
Definition (Operations on \(\mathbb R\)): For \([\alpha],[\beta]\in\mathbb R\) with \(\alpha=(a_{n})\) and \(\beta=(b_{n})\) define \[ [\alpha]\,+\,[\beta]\;:=\;[\, (a_{n}+b_{n})_{n\in\mathbb N} \,], \qquad [\alpha]\;\cdot\;[\beta]\;:=\;[\, (a_{n}b_{n})_{n\in\mathbb N} \,]. \]
Suppose \(\alpha\sim\alpha'\) and \(\beta\sim\beta'\). Then \((a_{n}-a'_{n})\) and \((b_{n}-b'_{n})\) are null. For addition we have \[ (a_{n}+b_{n})-(a'_{n}+b'_{n})=(a_{n}-a'_{n})+(b_{n}-b'_{n}), \] a sum of two null sequences, which is again null; hence \((a_{n}+b_{n})\sim(a'_{n}+b'_{n})\). For multiplication, \[ a_{n}b_{n}-a'_{n}b'_{n}=a_{n}b_{n}-a_{n}b'_{n}+a_{n}b'_{n}-a'_{n}b'_{n} =a_{n}(b_{n}-b'_{n})+b'_{n}(a_{n}-a'_{n}), \] and each term is a product of a bounded rational sequence with a null sequence, which is again null. Therefore the product classes are equal. ∎
Definition (Embedding of \(\mathbb Q\) into \(\mathbb R\)): For a rational \(q\in\mathbb Q\) consider the constant sequence \(\mathbf{c}_{q}=(q,q,q,\dots)\). Its class \([\mathbf{c}_{q}]\) is a real number. The map \(q\mapsto[\mathbf{c}_{q}]\) is an injective homomorphism, allowing us to identify \(\mathbb Q\) with a subset of \(\mathbb R\).
Definition (Absolute value on \(\mathbb R\)): For \([\alpha]\in\mathbb R\) with \(\alpha=(a_{n})\) define \[ \bigl|[\alpha]\bigr|:=\bigl[\,|a_{n}|\,\bigr] . \] This is well‑defined because \(|\cdot|\) on \(\mathbb Q\) preserves null differences.
With the operations and order induced from the rationals, \(\mathbb R\) satisfies the **completeness axiom**: every Cauchy sequence of real numbers converges to a real number. In our construction this holds by definition: a Cauchy sequence of reals is represented by a Cauchy sequence of Cauchy sequences of rationals; taking a diagonal subsequence yields a single Cauchy sequence of rationals whose class is the limit. Consequently, \(\mathbb R\) is a complete ordered field, which characterises the real numbers up to unique isomorphism.