User blog:Edwin Shade/Complete List Of Blog Posts (Part II)

Among the many paradoxes of mathematics, the one I have found to be the most thought-provoking is the idea that an infinity can be too big to count as an infinity, and by virtue of being too big for it's britches, be said not to exist. The classic reasoning is as follows: that if the set of all sets were a set, then the set would also contain it's own power-set, which is by virtue larger than the original set, and hence the set of all sets is not a set. This is true reasoning, but rather than proving the collection of all infinities, or sets, cannot exist in one formally defined 'bundle', all we have really proven is that a set is too small to contain all sets. Surely though, there is nothing past the notion of sets that we have to fear, something that might transcend current number theory - right ?

To begin with, let us ask ourselves the question: Does there exist a number that is the solution to the equation $$n=n+1$$ ? You will be forced to admit there is no such number. You however would absolutely agree with me if I claimed that the property $$n\neq n+1$$ is shared by all natural numbers. What then, if we denied the notion of infinity because it satisfies this equation, whereas we have established there is no numerical solution ? That would be silly, because while there is not a numerical solution, there is an infinite solution, $$\aleph_0$$, which is not a number, which need not conform to our rules about numbers, as it is not a number itself. This is virtually the same reasoning though as is employed in the proof there is not a collection of all sets. We make an assumption that the collection of all sets is a set, and in doing so we disprove it. This gets us nowhere however, for the collection of all sets clearly does exist in abstraction, for if the collection of all sets didn't exist, then neither would any set, and therefore no sets would exist. The issue here is that we must invoke a higher form of 'structure', such as we invoked the infinite to solve a predicament with the finite, to describe the entirety of a lesser type of structure. In the case of the natural numbers this higher structure is $$\aleph_0$$, which describes the entirety of the finite structures beneath it, namely, the set of natural numbers. It should come as no surprise then, that to speak properly of the collection of all sets, we must use a higher type of structure than a set, which can obey laws that would seem contradictory to the laws governing a set, but is nonetheless a valid conception.

Let the collection of all sets therefore, be denoted by $$\mho$$, or the Greek letter Omega turned upside-down. This symbol is most appropriate, as it's cauldron-esque shape reminds us that it contains all sets, and therefore is composed of a rich broth of contrasting infinities. $$\mho$$ is too big to be infinite, and therefore is neither finite nor transfinite, but falls into a class of it's own, which Cantor considered the absolutes. I have chosen to call this specific collection of sets The First Dominion. As $$\mho$$ is too large to be contained within a set it must be contained within a higher order structure I dub a soup, and is to be denoted with brackets and a subscript with a one, or $$\{\mho\}_1$$. Naturally the limit of all soups is the collection of all soups and First Dominions, which is be denoted $$\mho_1$$, or The Second Dominion. Just as we can construct The Second Dominion we can construct The Third Dominion by a similar means. I will go into more detail on this as time progresses.

Any question, comments, or ideas are fully welcome !