Cantor, Russell, and ZFC
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Cantor's research on sets and his creation of the continuum hypothesis, CH, in 1878, have become a perplexing problem for mathematicians with no complete and satisfactory solution having been accomplished. Some of the problems, which have emerged from the research conducted on sets, are the contradictions and the creation of paradoxes; more specifically, Cantor's paradox. As set theory began to evolve, another paradox surfaced, which was named Russell's paradox. This paradox stunned the world of Mathematicians, and has continued to be a problem to this day. The ZFC have produced axioms to address the issues caused by Russell's paradox, but sometimes these too have come up short. In this paper, two concepts are used, Ispace (the imagination) and Tspace (three dimensional reality, where all things real exist), in order to shed new light on the problems of set theory and the CH. This is a new method for solving these problems.
John Northern
Doctor of Chiropractic
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- Rating: 5 out of 5 stars5/5The writing is concise and can be easily followed and there are no rigorous proofs and concepts are explained - specifically works on solving Cantor's and Russell's paradox.
Book preview
Cantor, Russell, and ZFC - John Northern
Introduction
It is because of the concept and the use of infinity that much of the confusions, contradictions, and paradoxes have arisen in set theory.
Ideas are created in the imagination, and the Big 3 (scientists, philosophers, and mathematicians) research the ideas, and finally, if the ideas can become a part of reality, then the ideas enter the world of technology, where inventions are made. Since ideas and abstractions can only exist in Ispace, then much of the time, the Big 3, do their work in Ispace; however, some inventors do most of their work in Tspace with a hit or miss approach, such as Edison and the light bulb.
The idea and the use of Ispace and Tspace were successful in solving Zeno's paradoxes, because Ispace and Tspace separate the ideas in the imagination from what is real, and also because Tspace is only composed of space and matter, which greatly simplifies the tenets of the problems. By separating the ideas from the facts, there is a better understanding of the problem(s). Too often ideas are created in Ispace, and from those ideas, statements are made which cannot translate to Tspace. Nevertheless, the statements are made as if they are a part of reality, and this becomes an error, which can lead to illogical assumptions, conclusions, and paradoxes. This paper will address these issues, including some of the ideas and some of the facts of infinity.
In many situations, infinity is a difficult concept to understand and to work with for mathematicians, philosophers, and scientists. For centuries they have tried to broach this subject, but as it turned out, there was little success, and eventually they learned how to work around it instead of with it. But Tspace lays down strict rules for infinity, and this clears up a lot of the confusion created by infinite concepts in Ispace and Tspace.
The different sizes of infinity, brought about by Cantor's research on sets, and especially his work on integers and real numbers, can be more clearly understood through the use of Ispace and Tspace. There are an infinite number of classifications of matter, such as an infinite number of trees, or an infinite number of rose bushes, or people, or cats, . . . . (The idea that there can be an infinite number of real objects will be logically proven in a book named: Is, Ts, infinity, authored by John Northern (and coming out soon)). But for now, the crux of this paper lies in the infinite sets of numbers (in Ispace) and in the cardinality of sets, which is the basis of Cantor's continuum hypothesis and his paradox; and infinite sets, which is the basis of Russell's paradox.
In the past, the CH (continuum hypothesis) has been unsolvable, but this paper shows a new method, using Ispace and Tspace, for tackling the unsolvable problems. There is a process for dealing with these types of problems, even if they cannot be solved.
If those of the Big 3 are not concerned with relating to the real world, such as those who are involved in highly advanced theoretical mathematics, then it's not necessary for them to differentiate between Ispace and Tspace; however, if the philosophers, scientists, and mathematicians are concerned with discoveries that will lead to new inventions, for example, an antigravity motor, then during, or even before the research