Axioms

Equality Axioms

1. Reflexivity for Equality: For any real number $a$, $a=a$.
2. Symmetry for Equality: For any real numbers $a$ and $b$, if $a=b$, then $b=a$.
3. Transitivity for Equality: For any real numbers $a$, $b$, and $c$, if $a=b$ and $b=c$, then $a=c$.
4. Substitution (Axiom): For any real numbers $a$ and $b$, if $a=b$, then $a$ may be replaced by $b$ (or $b$ by $a$) in any mathematical proposition without changing the truth value of that proposition.

Addition Axioms

1. Closure for Addition: The set $\mathbb{R}$ is closed with respect to addition. If $a$ and $b$ are real numbers, then so is their sum $a+b$.
2. Commutativity for Addition: For every $a$ and $b$ in $\mathbb{R}$, $a+b=b+a$.
3. Associativity for Addition: For every $a$, $b$, and $c$ in $\mathbb{R}$, $(a+b)+c=a+(b+c)$.
4. Existence of an Additive Identity: There exists a unique real number, which is $0$, such that for every number $a$, $a+0=0+a=a$.
5. Existence of an Additive Inverse: There exists a unique real number, which is $-a$, such that for every number $a$, $a+(-a)=(-a)+a=0$.
6. Addition Property for Equality (APE): For every $a$, $b$, and $c$ in $\mathbb{R}$, if $a=b$, then $a+c=b+c$.

Multiplication Axioms

1. Closure for Multiplication: The set $\mathbb{R}$ is closed with respect to multiplication. If and are real numbers, then so is their product $a\ast b$.
2. Commutativity for Multiplication: For every $a$ and $b$ in $\mathbb{R}$, $a\ast b=b\ast a$.
3. Associativity for Multiplication: For every $a$, $b$, and $c$ in $\mathbb{R}$, $(ab)c=a(bc)$.
4. Existence of an Multiplicative Identity: There exists a unique real number, which is $1$, such that for every number $a$, $a\ast 1=1\ast a=a$.
5. Existence of an Multiplicative Inverse: There exists a unique real number, which is $\frac{1}{a}$, where $a\ne 0$, such that for every number $a$, $a\left(\frac{1}{a}\right)=\left(\frac{1}{a}\right)a=0$.
6. Multiplicative Property for Equality (MPE): For every $a$, $b$, and $c$ in R, if $a=b$, then $ac=bc$.

Distributive Axioms

1. Distributivity of Multiplication over Addition: For every a, b, and c in R, $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$.
2. Trichotomy: For any two real numbers a and b, one and only one of these three propositions are true: $a<b$, $a>b$, or $a=b$.
3. Transitivity for Inequality: For every a, b, and c in R, if $a<b$ and $b<c$ then $a<c$. Basically, $a<b<c$.
4. Addition Property for Inequality (API): For every $a$, $b$, and $c$ in $\mathbb{R}$, if $a<b$ and $a+c<b+c$.
5. Multiplication Property for Inequality (MPI): For every $a$, $b$, and $c$ in $\mathbb{R}$, if $a<b$ and $c>0$ and $a\ast c<b\ast c$.