Bertrand Russell's Paradox, and Set Theory

In my book "Everything is Something," I set about to accomplish many things, one of which was to refute Bertrand Russell's Paradox. After finishing the book, and publishing it, I realize that I could have phrased my account of his paradox differently, and more accurately. In the book, I describe his paradox as "a set that is not a member of itself." In retrospect, it would have been more accurate for me to have described it as "the set of all sets that are not members of themselves." His logic actually makes far more sense in the second description, than in the first, which I had said. However, in "Everything is Something," I prove, and demonstrate using logic, that every set is a member of itself. A set is the set of all of its members, therefore the set is equal to the set of all of its members, therefore the set is equal to all of its members, and all of its members are within the set, therefore the set is within the set, therefore the set is a member of itself, and this is true of all sets, because it is true of a set as such, of the essential set, because of the essence of a set, and is deduced without reference to any specifics that might differentiate this set from any possible set, which is the methodology of essential logic. The fact that every set is a member of itself refutes both formulations of Bertrand Russell's Paradox, so my error was only in my statement of his paradox, there was no error in my statement of my proof of my refutation of his paradox. The only set which is not a member of itself is the set with no members, which is the null set, the empty set, but, because it has no members, the null set does not exist, it is equal to zero, because it contains zero members, it contains nothing. It can be thought about and conceptualized, but it does not, and can never, exist, in the sense of being a real thing, in reality.

More broadly, regarding set theory, which I discuss at length in "Everything is Something," I later realized this, also: Bertrand Russell (and the early Ludwig Wittgenstein) sought to derive math from set theory, and to prove math using set theory. But math is set theory, math and set theory are identical, so their effort, their quest, their work, was doomed to failure from the very beginning. For example, what is the difference between "a set of ten things," as such, and "the number ten," as such? What is the difference between "a set of one thing," "a thing," and "the number one"? In my opinion, there is no difference. So, the attempt to derive math from set theory, and to prove math using set theory, reduces to the attempt to derive math from math, and to prove math using math, which adds nothing substantive to math itself, as an academic discipline. Philosophy can say how we gain knowledge of math, and what role math plays in the world, and what math is useful for, but math is math, and philosophy does not, fundamentally, add anything to the statement "math is math," for one does math by doing math, not by analyzing math by means of philosophy, although philosophy can add to the understanding of math, to our wisdom with respect to math, so to speak, for example by saying whether numbers exist in the physical world or exist in a separate spiritual or intellectual world or do not really exist at all, which is within the province of philosophy.

And I argue that numbers do physically exist, and there is no need to refer to or rely upon another dimension or an intellectual or spiritual world where numbers exist, and it is also obviously not true to say that numbers do not exist or they are merely our way of speaking about things but they themselves are not real, because we can see that numbers are real, if we look out the window and see a flock of ten robins then we can see with our own eyes, and know beyond doubt, that the number ten is real, because a thing is a set of properties, and an essence is one property or set of properties isolated out of a thing or things and then analyzed using logic, and a number is the essence of a group of things as being a certain particular number or amount or quantity of things, so, if groups physically exist, if, for example, there is a group of ten squirrels living in my back yard, and they are real, then the number ten physically exists, because the group of ten squirrels is a thing that has a property of being the number ten, and a property of being a group of squirrels, and a property of being in my back yard, and the property of being alive today, and so on, for example, and "the number ten" simply uses essential logic to isolate the property of being ten, from a real physical group of ten things, or from many such groups, which is abstraction and induction, and then uses the number ten in math to prove things in math, which is deduction and logic. For another example, if there are three boxes on my shelf, those boxes are the number three, plus many other properties such as being boxes and being on my shelf, and the number three physically exists as those three boxes, because that real thing has the property of being three. And there is no difference between that number ten, as a group of ten things, and that number three, as a group of three things, on the one hand, and the numbers ten and three of abstract theoretical mathematics, in intellectual math, on the other hand.

Our knowledge of math does come from the physical world, and this is why math, and math used in science, can describe the physical world, and is useful for technology in physical reality. As I have also written elsewhere, I believe that math works in science, and math describes the physical world, because I do not believe that there is any real difference between the theoretical abstract space and time of math, on the one hand, and our actual real physical space-time, on the other hand, the only difference is that the first space-time is thought about in some mathematician's mind, and the second space-time is one which exists objectively, outside of our minds, and which we experience as the physical world, but the actual math is exactly the same for both. So, when mathematicians think of an intellectual or spiritual world where math is real, it is actually our own physical reality that they are thinking about. Science, and the scientific world, is essentially the manifestation of the abstract mathematics of space-time into the concrete experiences that you have as a person observing and experiencing the little slice of space-time in reality that your point of view is privileged to experience. Science is really just how all the math fits together to form reality, and to form our experience of reality.