Saturday, September 28, 2013

Mathematical ions

The most common numbers used in mathematics have an ordering relation <= associated with them for example the natural numbers appear in the order 0,1,2,3,4,... the integers appear in the unbounded order ...,-2,-1,0,1,2,... the rational numbers are are densely ordered variant of the integers and the reals are the dedekind completion of the rational numbers.

The ordinal numbers are the order types of well orders and therefore they include the natural numbers 0,1,2,3,4,... mentioned before as well as infinite numbers such as $\omega$, $2\omega$, $\omega^2$ and $\omega^\omega$. The surreal numbers are a large ordered field that include the natural numbers, the integers, the rational numbers, the real numbers, the transfinite ordinals, infinitesimals, and much more.

It is my contention that numbers should be related to ordering and not just arithmetic so the so called "complex numbers" belong to the class of ions which includes binarions, quaternions, octonions, and sedenions rather then the class of numbers. Numbers should be related to ordering which implies that the class of surreal numbers is the single class of numbers to rule them all. In conclusion every number is a surreal number.

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