Part 2
At the time of Cauchy and Gauss in the middle of 19th century, the plurality of those who were teaching mathematics grew rapidly. Why did Abel see mathematics in a mess? Abel meant its lacking rigorous foundation. While the irrational numbers were well known to be different from the rational ones, it was common practice to ignore this trifle. When Riemann suggested his surfaces already in 1851, he referred to points, not to measures. After Cauchy had defined irrational numbers as limit values attributed to convergent series, it was and it is still tempting to interpret limit-values as limit-points instead of limit-measures and furthermore to logically equate the never-ending procedure that defines a series of quantities with the limit attributed to it.
Among a lot of pre-thinkers of Dedekind and Cantor were not just rather self-taught ones like Bolzano and Dirichlet but also Heine and the utterly influential teacher Weierstrass. Even Cantor's strongest opponent Kronecker shared their intention to make the continuum algebraic.
Dedekind declared the real numbers an extension of the rational ones. Actually we have to interpret the transition from rational to real as an essential extension, as the transition from discrete to continuous, i.e. from quantity to quality. In other words, in contrast to algebraic numbers, real numbers are not numerically distinguished from each other. For instance, 3.14... is a rational number, no matter how many subsequent decimals. Its limit-measure pi is qualitatively different.
Dedekind made two unwarranted claims:
- He postulated that a given irrational number can be subject to the distinction between equal to, larger, or smaller than a given rational number. He admitted; nobody can provide evidence for that.
- He imagined the real line composed of just enough densely located points. While this contradicts to the definition of a point as having no parts and the continuum of real numbers as endlessly divisible, this idea of Dedekind can be rescued by replacing points with infinitesimal measures.
G. Cantor started with the naive idea that there must be more real than rational numbers and spoke of more than countable many (überabzählbar viele) alias transfinite numbers. As did already Galileo he used the method of bijection in order to demonstrate that the infinite amount of natural numbers is sufficient as to count any rational or algebraic irrational constructs. The latter was a surprise that paved the way for publishing his otherwise silly idea despite of rejection by Kronecker already in 1874. While Galileo had plausibly concluded that there are not more natural numbers that their squares because a quantitative comparison is not reasonable for infinite series, Cantor and Dedekind followed the intuition that only counting means (well) ordering from smaller to larger, a view that denies actual infinite non-linear divisibility. Cantor's proofs rested on the assumption that any number must be smaller than, equal to, or larger than a given rational one. Ignoring the possibility of incomparability Cantor's evidence is logically cyclic. Up to now, mathematicians tend to fail understanding what makes real numbers different from rational ones, cf. an admission by Ebbinghaus in his textbook "Numbers".
Fraenkel 1923 reveals the decisive trick in Cantor's 1891 diagonal argument: Cantor assumed that all of infinitely many digits are fixed. Likewise one could fix all of the natural numbers and then show that by adding something that there are evidently more than infinitely many natural numbers: "Infinity plus anything is larger than infinity" ?
Already the word "many" instead of "much of" is misleading: In common sense, an endless amount is not countable. Cantor's "mathematical" notion of being countable contradicts to the traditional understanding. The entity of all natural numbers is not countable. As Archimedes understood, numbers are endless. They are just tools. The allegedly difficult to understand Cantor's Set Theory is easily understand as a useless intentional denial of sound Galilean reasoning.
What was the intention? Why declared the Bourbakis Set Theory the fundamental of mathematics? See above.
Eckard