Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space) . Also, set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinity of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor’s work was the subject of much criticism to the effect that it dealt with fictions—indeed, that it encroached on the domain of philosophersand violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and by the 1890s Cantor’s ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics.

Dear Dr. Boushra Youssif, thank you.

The mutual contradiction in nature and the confusion in form of “potential infinite, actual infinite” in the foundation of present classical infinite relate science system have been making people unable to know what kind of X--->0 number form is potential infinitesimal and what is actual infinitesimal and, unable to know how to treat such different things with different natures in mathematical analysis. So, following two insurmountable obstacles in the process of “quantitative cognition to X--->0 number form” have been unavoidable constructed:

(1) Theoretically: on the one hand, it is impossible to know any of the X--->0 number forms treated in front of us are potential infinitesimal or actual infinitesimal; on the other hand, as long as any infinite related numbers with the form of X--->0 appear in mathematical analysis, they are asserted as the same some “non-number numbers (variables) of both potential infinitesimal and actual infinitesimal”, and there is no systematic qualitative and quantitative analysis and processing theory at all for the “ X--->0 potential infinitesimal or actual infinitesimal number forms”.

(2) Operationally, when facing “X--->0 number forms” with different identities in operations, no one knows when they should be potential infinitesimals and when they should be actual infinitesimals and parroting is the only way. The contemporary mathematics version of Hans Christian Andersen’s fairy tale "the emperor's new clothes" has been seeing frequently in present classical mathematical analysis------- it is ok as long as many people say that the emperor has new clothes on, it doesn't matter what the truth is. There is no systematic quantitative analysis and processing theory at all for the “ X--->0 potential infinitesimal or actual infinitesimal number forms”! So, for any infinite related numbers with the form of X--->0 (whether "potential infinitesimal or actual infinitesimal" or whether "could make it zero, take its limit, take the standard number" ), shall be treated in accordance with the specifications laid down by some traditional piplines (this is the very reason why three generations of mathematical analysis completely equivalent to each other). For example, in the quantitative cognizing operations relating to any "unknowing--what X--->0 number forms”, people can do freely in any of following different pipelines------the first pipeline is: "not to make them zero, take their limit, take the standard numbers, …" at the beginning and during most of the operating process but "make them zero, take their limit, take their standard numbers, …" at the final step; the second pipeline is: "never make them zero, take their limit, take the standard numbers, …" from beginning to the end during the whole operating process; ….

The newly discovered Harmonic Series Paradox is a typical example: the confusion of “potential infinite, actual infinite” contents in present classical mathematical analysis resulted by the confusion of “potential infinite, actual infinite” concepts in present classical infinite theory system have been making people unable (impossible) to know what those “X--->0 number forms (items)” are in newly discovered Harmonic Series Paradox and even more, impossible to know whether they are potential infinitesimals or actual infinitesimals and how to treat them. Studies proved that the so called “infinitesimal” in present classical mathematical analysis are in fact a mixture of two “infinite mathematical carries (number forms)”: “X--->0 intersmall” and “X--->0 infinitesimal” [9-19]. In many practice, one cannot distinguish at all the two different kinds of "mathematical carrier" and can only abuse of the limit theory operationally and theatrically-------it is wrongful (unscientific) to treat abusely "two number forms of Un--->0 with different natures" in Harmonic Series in a pipeline operation of "bracketing” by limit theory to create infinite numbers greater than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... and turn the Un--->0 Harmonic Series into a “Vn ---> any positive constants” infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000). The newly discovered Harmonic Series Paradox has become a typical contemporary mathematical version of Andersen’s fairy tale "the emperor's new clothes" and, the suspended Ancient Zeno’s challenging statement of “not matter how fast Achilles runs, he will never catch up with the slowly crawling Turtle in Achilles--Turtle Race” has been confirmed with exactly the same operations and same ideas basing on modern limit theory as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Achilles--Turtle Race Theorem but not Suspended Zeno’s Achilles--Turtle Race Paradox.

So, some members of Zeno’s Paradox Family are taken as paradoxes while some are taken as “great theorems” in our present infinite related mathematics.

Yous,

Geng Ouyang

@ Boushra Youssif

(Number) `i´ shows that you don´t get a solution for the exercise to take the square-root!

Reform could!

Set theory wants to categorize infinite (quality) sets by steps of more elements in number (of amount). But the difference shall come by quantity!

Reform shows: each infinite set is equal in quality one to another. IN –> IQ –> IR exists!

Quantity never beats quality.

See also: Pietro Dell´Acqua, Could I have comments about my ”Note on Cantor´s diagonal argument”?

Thank you for your frank and kind idears Mr. Marcel Van de Vel,

I agree with you that my work has a lot to do with philosophy and this is one of the main reasons many purely mathematical workers do not agree with my ideas on philosophical--mathematical issue because their work has nothing to do with foundation of mathematics.

1, do you think the 2500 years suspended Zeno’s Achilles--Turtle Race Paradox is a purely philosophical issue?

2, do you think the modern version of Zeno’s Achilles--Turtle Race Paradox (newly discovered Harmonic Series Paradox in mathematics) is also a purely philosophical issue?

---------with the operation of "bracketing” by limit theory to create infinite numbers greater than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from those Un--->0 items in Harmonic Series and turn the Un--->0 Harmonic Series into a “Vn ---> any positive constants” infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000).

3, the paradox families in present mathematical analysis and set theory (infinite paradoxes symptom complex) drew me to the foundation of mathematics, and the “‘potential infinite--actual infinite’ confusing” abyss has been detected: not only 2500--year Zeno’s Achilles--Turtle Race Paradox can not be solved but George Berkeley’s Paradox for The Second Mathematical Crisis and The Russell’s Paradox for The Third Mathematical Crisis are unsolvable at all in present infinite related science and mathematics basing on classical infinite theory system with its “confusing ‘potential infinite--actual infinite’” and, I am very sure that more paradoxes by the “‘potential infinite--actual infinite’ confusing” will be produced if the idea of “potential infinite--actual infinite” are still with humans science and mathematics.

My conclusion is: Standard Analysis, Nonstandard Analysis, Type Theory, Model Theory, ZFC did nothing at all to help us jump out of the “‘potential infinite--actual infinite’ confusing” abyss.

Thank you again.

Yous,

Geng Ouyang

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

2，On the Quantitative Cognitions to “Infinite Things” (II)

3，On the Quantitative Cognitions to “Infinite Things” (III)

4 On the Quantitative Cognitions to “Infinite Things” (IV)

5 On the Quantitative Cognitions to “Infinite Things” (V)