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This book explains why your introductory Real Analysis textbook may not make complete sense to you and why mathematics is so counterintuitive. It lists the seeming contradictions in mathematics and identifies the root cause of all counterintuitive results. It points out potential inconsistencies in Cantor’s theorem and in the concept of actual infinity. The definition of the infinite set of natural numbers N violates the concepts of a set and of infinity. It proposes to physicists that their unsuccessful attempts at a unified theory of physics may be due to the underlying mathematics. The first course in Real Analysis comes as a rude shock to most students. How can mathematics suddenly become so counterintuitive and difficult to comprehend? The introductory Real Analysis textbook does not make complete sense. Parts of Real Analysis consist of imaginary concepts without any “real” examples. One would expect Real Analysis deals with numbers such as 3.1 and 4.267 and the like. It turns out in the end almost all real numbers are irrational and they must be denoted with an infinite non-repetitive decimal representation. Infinity is not a number and real numbers are defined in terms of something that is not a real number. In fact, not a single irrational number can be written in the decimal notation as it is not possible to lay out infinite non-repeating decimal digits. The irrational numbers are pretty much imaginary. How strange? The course should be titled “Unreal Analysis”. Modern mathematics is based on Cantor’s infinite set theory which is full of counterintuitive results and seeming contradictions. This book raises doubts about many results presented in a course on Real Analysis and describes the various ways the results seem to be inconsistent and often contradictory. The “smoking gun” that something is amiss is the theorem on the density of the rational numbers which is built on an even more elementary concept – the Archimedean Property. How can a countable set of rational numbers be densely distributed inside an uncountable set of real numbers representing a higher order of infinity? No text book explains how this is possible. The only other science where such abstractions come directly into play is physics. A different theory of mathematics may lead to different theories in physics. The unsuccessful attempts at a unified theory of physics may be due to the underlying mathematics. Certain conclusions about imperfections in the real small world of quantum mechanics, like the Heisenberg Principle, may instead be pointing to an imperfection in the underlying mathematics.
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