Cantor-Bernstein–Schroeder theorem

``If each of two sets A and B are equivalent to a subset of the other, then A is equivalent to B.

This is a vital result in the study of cardinal numbers, indeed a vital result in the development of set theory. When Cantor saw Bernstein's proof he was so impressed that he communicated it to Emile Borel and it was published in Borel's Lecons sur la theorie des fonctions in 1898. We mentioned above that the result is known as the Schröder-Bernstein Theorem. Ernst Schröder independently published a proof in 1898 but Schröder's proof was later shown to contain an error although the basic idea can be corrected to give a proof. In fact the theorem was stated by Cantor in Beiträge zur Begründung der transfiniten Mengenlehre but his justification of the result there is not a rigorous proof (as he himself was aware) and, as we explained above, it was while correcting the proofs of this work that Bernstein, still a high school student, constructed a correct proof. In 1905 Bernstein published another important article on transfinite ordinal numbers Über die Reihe der Transfiniten Ordnungszahlen which appeared in Mathematische Annalen.  ''

Taken from The MacTutor History of Mathematics archive

My son's name is Lev Cantoral Bronstein.

Neat.

Felix Bernstein also contributed to understand the genetic origin of blood types, from the same source:

``However, it was not until Berstein's two papers Ergebnisse einer biostatistischen zusammenfassenden Betrachtung über die erblichen Blutstructuren des Manschen (1924) and Zusammenfassende Betrachtungen über die erblichen Blutstructuren des Manschen (1925) that the correct hereditary mechanism was proposed [2]:-

The correct hypothesis of multiple alleles at one locus was not demonstrated until Bernstein did so in 1924 and 1925.

Crow asks:-

Why was the earlier incorrect hypothesis so widely accepted from 1910 to 1924?

The reason seems to relate to errors in the data being used, partly due to misprints in the published data, but also, perhaps not surprisingly, due to the fact that a child's biological father may not be the husband of the child's mother (often unknown to the husband).''