This Theorem is Pretty Dope

Now let’s get onto what all I’ve done in the field of mathematics. In short, I’m basically the founder of set theory, and I’ll tell you why. At the time (i.e. before my work to begin the study of set theory) the idea of a set was pretty simple and used as a sort of implicit and elementary basis, dating back all the way to stuff like what Aristotle came up with, so it was thought of as a sort of trivial concept along with its contents. Moreover, there were finite sets, and this abstractness of “the infinite” which was more of a philosophical topic than mathematical. However, I’m one of the dudes who showed that set theory isn’t just a trivial matter by shedding more light on some of its topics. In my first paper in 1874, I provided the first theorems of transfinite sets. A huge part of this is about these developed concepts of one-to-one correspondences, “countability” (having to do with cardinality), density (of sequences), and (probably most divisively) the existence of an infinite amount of sets having different infinite cardinalities. For example, we know it is common knowledge that real numbers, natural numbers, etc. are infinite. But, it was not common knowledge that the reals were the same size (cardinality, or think of it as “just as numerous") as the irrationals, and both of which are larger than the size of the natural numbers. Basically, that there are different sizes (moreover, an infinite amount of different sizes) of infinity, which I call transfinite. Transfinite numbers are, as previously stated, infinite, yet they can be increasable in magnitude, unlike the Absolute infinite which displays supreme perfection, independency, extra-worldly existence, and admits no kind of determination like that which is in God.

But wait! There’s more! This was just the start. As I worked further with set theory, I helped start the fundamental constructions of set theory like power sets, more detailed ideas about subsets, cardinality, ordinality, and the like; much of which is introduced in my first paper, and refined and elaborated on in subsequent works. For example, I created another different proof of the uncountability of the real numbers in my 1891 article using what I call the diagonal argument, and I have also built upon Liouville’s construction of transcendental numbers by finding a new method to construct them, thus also providing a different proof of his theorem that every interval contains infinitely many transcendental numbers. 

I feel like you would (or at the least I think you should) be wanting of some sort of mathematics, so here I will resolve your desires with a proof of my theorem that the reals are uncountable (i.e. the cardinality of the reals exceed that of the natural numbers, the set which provides the means of counting). I provide an image to visually understand and for you to figure out the more famous diagonal argument:

but here show a proof by contradiction:

Suppose, for contradiction, that there exists a sequence {xn} such that for all y in R, there exists an n such that xn = y (thus supposing that there does exist a countable sequence containing all real numbers). Construct a sequence In of closed, bounded, non-empty intervals (say that interval is [a,b]) such that In ≥ I(n+1) and xn is not in In (that is, make a sequence of nested intervals which does not contain the n’th real number. The way of doing this is that if x(n+1) is in the first/lower half of the n’th interval, choose your next interval to be the last quarter of your original interval (i.e. if x(n+1) ≤ (a+b)/2) then take I(n+1) to be [a+3/4*(b-a), b]). But if it is in the second/upper half of the interval, choose the next interval to be that first quarter ([a, a+1/4(b-a)]). This way you are guaranteed that your next term is not in your next interval (i.e. x(n+1) is not an element of I(n+1) ≤ In). Now keep constructing more and more of these nesting intervals by taking the part that definitely does not have the next real number in it. However, by the nested interval theorem, the intersection of all of these infinite intervals is non-empty. That means that you can choose some y in that intersection which is not any real number xn, which is a contradiction, meaning R is uncountable. []

I realize this post was very optimistic and about many of the gifts I have provided the mathematics community. As such, since that is not the whole story, my next post shall include a little bit of the push-back and issues that arose alongside my genius.