How did Cantor find the overrun?

as you know, the overrun number ( transfinite ) The core concept of set theory. We know that apart from the problem of how the limit number is found, it is essentially a matter of figuring out how the set theory was created.

in three-dimensional space, the origin of coordinates O the Cartesian coordinates are ( 0,0,0 ). Thus, we can infer that in the infinite dimensional space, there are similar cases. Looking back on the history of Europe, the Triangle series is a very popular research area. However, the key problem of Triangle series convergence is the uniqueness of the limit function, if the uniqueness of the convergence of trigonometric series cannot prove, the whole modern mathematics must be messy.

in the circumstances, many well-known mathematicians knew that the problem was a "nut", prohibitive. However, in the teacher's advice, in 1870 years, only 20 of the "Little head" Cantor but chewed down this piece of "nut"! His outstanding achievements are now in the annals of modern mathematics development.

The Cantor theorem for trigonometric series is: if the SUM function of trigonometric series f (x) is zero, then the coefficients of the trigonometric series are zero. Thus, the convergence uniqueness of the series is solved.

Further, Cantor envisages that trigonometric series and function values are not all zeros, even, and the function of "non-0 points" is infinitely infinite, however, its "limit point" only a limited number, at this time the uniqueness of the convergence of trigonometric series can still be established. In the process of proving, Cantor found that the subscript of some series items needed to be expressed by "super Limit number", thus, Cantor initiated the idea of introducing "super Limit number".

What we need to understand is that the establishment of modern set theory is not a personal "inspiration flash" of Cantor, but is caused by the internal demand of mathematical development. Cantor is not a man of God.

Description: As the name implies, the overrun number is the number that exceeds all the finite number, and there are other subsequent numbers behind it. The overrun number is also "number", not something else.

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How did Cantor find the overrun?