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\begin{center} \textbf{Axioms of the real numbers $\mathbb{R}$.} \end{center}


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\textbf{Axioms on addition:}

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Axiom A1 (closure property of addition).  If each of $x$ and $y$ is
a number then $x+y$ is a number.

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Axiom A2 (associative property of addition).  If each of $x$, $y$
and $z$ is a number then $(x+y)+ z = x + (y+z)$.

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Axiom A3 (commutative property of addition).  If each of $x$ and $y$
is a number then $x+y = y+ x$.

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Axiom A4 (identity for addition).  There exists a number $0$ so that
if $x$ is a number then $0+x = x$.

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Axiom A5 (inverses for addition).  If $x$ is a number, there is a
number denoted by $-x$ so that $x + (-x) = 0$.

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\textbf{Axioms on multiplication:}

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Axiom M1 (closure property of multiplication).  If each of $x$ and
$y$ is a number then $x \cdot y$ is a number.

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Axiom M2 (associative property of multiplication).  If each of $x$,
$y$ and $z$ is a number then $(x\cdot y)\cdot z = x \cdot (y \cdot
z)$.

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Axiom M3 (commutative property of multiplication).  If each of $x$
and $y$ is a number then $x \cdot y = y \cdot x$.

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Axiom M4 (identity for multiplication).  There exists a number $1$
so that if $x$ is a number then $1 \cdot x = x$.

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Axiom M5.  $0 \neq 1$.

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Axiom M6 (inverses for multiplication).  If $x$ is a number and $x
\neq 0$, there is a number denoted by $\frac 1x $ so that $x \cdot
\frac 1x  = 1$.

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\textbf{Relationship between addition and multiplication:}

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Axiom D (distributive axiom).  If each of $x$, $y$ and $z$ is a
number then $x \cdot (y+z) = (x \cdot y) + (x \cdot z)$.

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\textbf{Order axioms.}

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There exists an order relation ``$<$'' so that:

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Axiom O1. If each of $x$ and $y$ is a number then exactly one of the
following is true:

\qquad $x<y$, \qquad $x=y$, \qquad or $y<x$.

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Axiom O2. If each of $x$, $y$ and $z$ is a number $x<y$ and $y<z$,
then $x<z$.

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Axiom O3. $0<1$.

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Axiom O4. If each of $x$, $y$ and $z$ is a number and $x<y$, then $x
+ z < y + z$.

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Axiom O5.  If each of $x$, $y$ and $z$ is a number $x<y$ and $0<z$,
then $x \cdot z < y \cdot z$.

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Definition.  The integer $p$ is said to be positive if and only if $0<p$; $\mathbb{N} = \{ n \in \mathbb{Z} | 0 < n \}$.

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\textbf{Induction Axiom.}  Note: We need to establish the relationship between the integers and the reals.  To do that we define $0$ and $1$ to be integers and inductively (using the induction axiom) to define the positive integers $\mathbb{N}$ by: if $n \in \mathbb{N}$ then $n+1 \in \mathbb{N}$.  Then the integers $\mathbb{Z}$ consists of $0$ together with $\mathbb{N}$ together with the additive inverses of elements of $\mathbb{N}$.

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Axiom I. Suppose that $S$ is a subset of $\mathbb{N}$ containing $1$ such
that if $n \in S$ then $n+1 \in S$. Then $S=\mathbb{N}$.

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Theorem. The induction axiom is equivalent to the following: Every non-empty subset of $\mathbb{N}$ has a least element.

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\textbf{Completeness Axiom.}  Definition: if $M$ is a subset of the real numbers then  $B$ is an upper bound for $M$ means that $x \le B$ for all $x \in M$; $B$ is said to a least upper bound of $M$ if $B$ is an upper bound of $M$ and no element less than $B$ is an upper bound of $M$.

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Axiom C.  If $M$ is a subset of the reals $\mathbb{R}$ and there is an upper bound of $M$ then there is a least upper bound of $M$.

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Axiom C is equivalent to the following, called the Dedekind cut axiom.

Dedekind cut axiom:  If $L$ and $R$ are two non-empty subsets of the reals whose union is the reals so that every point of $L$ is to the left of every point of $R$ (equivalently, every point of $R$ is to the right of every point of $L$), then either $L$ has a rightmost point or $R$ has a leftmost point.

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\noindent Comments:

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The addition axioms (A1 - A5) hold for $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$.

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The multiplication axiom M1 - M6 hold for $\mathbb{Q}, \mathbb{R}$ but not for $\mathbb{Z}$ because to get multiplicative inverses you need to extend the integers to at least the rationals $\mathbb{Q}$.

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The rationals satisfy all the axioms except the completeness axiom.  You need the completeness axiom to prove that things like $\sqrt{2}$ exist.  If we have time I'll outline a proof later in the semester.  The complex numbers $\mathbb{C}$ don't satisfy the order axioms, but otherwise satisfy the Addition and Multiplication axioms.

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You may observe that axiom M5 follows from axioms O1 and O3.  The reason it's there is that the number system is typically built up in stages so that a bunch of theorems based on the addition and multiplication axioms can be derived before the order property is introduced, but the condition $0 \ne 1$ is needed for these theorems.  Also, (as an aside), one can define number systems that don't have orders (like the complex numbers or the complex `integers') that still need this axiom.

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