## Idea ## An axiomatization of the real numbers very similar to Tarski's axiomatization of the real numbers, in that it only uses a strict order $\lt$, addition $+$, and one $1$ to define the real numbers. ## Definition ## Strict order axioms: $\lt$ * For all terms $a:\mathbb{R}$, $a \lt a$ is false. * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, $c:\mathbb{R}$, $a \lt c$ implies $a \lt b$ or $b \lt c$ * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, not $a \lt b$ and not $b \lt a$ implies $a = b$. * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, $a \lt b$ implies not $b \lt a$ Denseness and Dedekind completeness axioms: [[open interval]]s, [[lower bounded open interval]]s, and [[upper bounded open interval]]s can be defined in any strictly ordered set * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, the open interval $(a,b)$ is inhabited. * For all terms $a:\mathbb{R}$, the lower bounded open interval $(a,\infty)$ is inhabited. * For all terms $a:\mathbb{R}$, the upper bounded open interval $(-\infty,a)$ is inhabited. * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, $a \lt b$ if and only if $(b,\infty)$ is a subinterval of $(a,\infty)$ * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, $b \lt a$ if and only if $(-\infty,b)$ is a subinterval of $(-\infty,a)$ * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, if $a \lt b$, then $F$ is a subinterval of the union of $(a, \infty)$ and $(-\infty, b)$ * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, the intersection of $(a,\infty)$ and $(-\infty,b)$ is a subinterval of $(a,b)$ Addition axioms: $+$ * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, $a + b = b + a$ * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, and $c:\mathbb{R}$, $a + (b + c) = (a + b) + c$ * For all terms $a:\mathbb{R}$ and $b:\mathbb{R}$, there exists a term $c:\mathbb{R}$ such that $a + b + c = a$ * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, and $c:\mathbb{R}$, $a \lt a + b$ and $a \lt a + c$ implies $a \lt a + b + c$ Archimedean property: * For all terms $a:\mathbb{R}$, $b:\mathbb{R}$, and $c:\mathbb{R}$, $a \lt a + b$ and $a \lt a + c$ implies that there exists a natural number $n:\mathbb{N}$ such that $b \lt n c$, where $n c$ is the additive $n$-th power (n-fold addition) One axioms: $1$ * $1 \lt 1 + 1$ ## See also ## * [[Dedekind real numbers]] ## References ## * Alfred, Tarski (24 March 1994), Introduction to Logic and to the Methodology of Deductive Sciences, Oxford University Press, ISBN 978-0-19-504472-0