User blog:Edwin Shade/Numbers As Big As Infinity

We often work with numbers such as $$\pi$$, $$e$$, and $$\frac{1}{3}$$, numbers whose representation in full requires an infinite number of digits. It bothered me slightly though, that although $$0.333...$$ is seen as a perfectly normal number, a number like $$333...333$$ is not accepted to be a proper number, because without a decimal point it would be infinitely large.

Perhaps there can exist a type of infinity more specific than $$\aleph_0$$, which has both leading and trailing digits on either end, but which extends forever in it's decimal expansion. For instance, $$10000000...0000000$$ could represent $$10^{\infty}$$, as the logical limit of the sequence $$\{1, 10, 100, 1000, 10000, 100000, 1000000,...\}$$. There arise some paradoxes similar to Thomson's lamp however when we try to calculate expressions such as $$-1^{\infty}$$, but only since we are using the $$\infty$$ symbol, which is rather vague. If we instead use an infinitely large number like $$1255082...1655361$$ then we know $$-1^{\1255082...1655361}=-1$$, because $$1255082...1655361$$ is an odd number...well, sort of. You cannot consider $$1255082...1655361$$ to be an odd number because odd numbers belong to the class of naturals, which are finite. Therefore we must treat $$1255082...1655361$$ as an odd number belonging to a group of natural numbers that are infinite in length. It can be seen that arithmetical operations on these infinitely long numbers are valid, as long as everything is explicitly defined enough so that an answer can be arrived at, and not two contradictory answers, forming a paradox.

This idea may be silly, but I thought I'd mention it because I've had the idea for a while now, and I'd like to see something come of it.