axioms of the real numbers
Axioms of the real numbers
These are the basic building blocks from which all theorems are proved.
We assume that there is a real number system, a set \[\mathbb{R}\] that contains two distinct elements 0 and 1 and on which are defined two binary operations (addition, \[+\] and multiplication, \[\cdot\]). We define addition and multiplication as \[+:\mathbb{R}\times \mathbb{R}\to \mathbb{R}\] and \[\cdot :\mathbb{R}\times \mathbb{R}\to \mathbb{R}\]. Note that the times symbol in \[\mathbb{R}\times \mathbb{R}\] refers to the Cartesian product. Then, the sum \[+(x,y)\] is written \[x+y\] and the product \[\cdot(x,y)\] is written \[x\cdot y\] or simply \[xy\]. Moreover, the following axioms must also be satisfied.
Field axioms for \[(\mathbb{R}, +,\cdot)\]
The real number system, with its distinct 0 and 1 and with its addition and multiplication is assumed to satisfy the following set of axioms, i.e. \[\mathbb{R}\] is a field under addition and multiplication. For all \[x,y,z\in \mathbb{R}\],
- Associativity of addition and multiplication: \[(x+y)+z=x+(y+z)\] and \[(x\cdot y)\cdot z=x\cdot (y\cdot z)\]
- Commutativity of addition and multiplication: \[x+y=y+x\] and \[x\cdot y=y\cdot x\].
- Distributivity of multiplication over addition: \[x\cdot (y+z)=x\cdot y+x\cdot z\].
- Existence of additive identity: \[\exists\, 0:x+0=x=0+x\], where \[\exists \,0:\] refers to "there exists \[0\in \mathbb{R}\] such that".
- Existence of multiplicative identity: \[\exists\,1\ne0:x\cdot 1=x=1\cdot x\], similarly \[\exists\,1\ne0:\] refers to "there exists \[1\in \mathbb{R}\] such that \[1\ne 0\]".
- Existence of additive inverses: For all \[x\] there exists \[-x\in \mathbb{R}\] such that \[x+(-x)=0\], symbolically this would be \[\forall\,x\in \mathbb{R},\exists\,x^{\prime}\in \mathbb{R}:(x+x^{\prime}=0)\].
- Existence of multiplicative inverses: For all \[x\in \mathbb{R}\,\backslash\left\{ 0 \right\}\] there exists \[x^{-1}\in \mathbb{R}\] such that \[xx^{-1}=1\], where \[x\in \mathbb{R}\,\backslash\left\{ 0 \right\}=\left\{ x\in \mathbb{R}:a\notin \left\{ 0 \right\} \right\}\], and symbolically this would be \[\forall\,x\in \left( \mathbb{R}\,\backslash \left\{ 0 \right\} \right),\exists\,x^{-1}\in \mathbb{R}:xx^{-1}=1\].
Order axioms
Assuming \[\mathbb{R}\] is an ordered field, i.e. there is a subset \[\mathbb{R}^{+}\] of \[\mathbb{R}\] where for all \[x,y\in \mathbb{R}\] the following axioms hold:
- Trichotomy: Exactly one of the following conditions holds: \[x\in \mathbb{R}^{+},\,-x\in \mathbb{R}^{+},\,x=0\].
- Closure of positive numbers under addition: If \[x\in \mathbb{R}^{+}\] and \[y\in \mathbb{R}^{+}\] then \[x+y\in \mathbb{R}^{+}\].
- Closure of positive numbers under multiplication: If \[x\in \mathbb{R}^{+}\] and \[y\in \mathbb{R}^{+}\] then \[xy\in \mathbb{R}^{+}\].