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[HM] Did Cantor prove the Schroeder-Bernstein theorem?
Posted:
Mar 26, 1999 11:18 PM
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In "The beginnings of set theory"
(see http://www-groups.dcs.st-and.ac.uk/~history/HistoryTopics.html)
it is said that Cantor, in his final double treatise (1895-97),
"proves that if A and B are sets with A equivalent to a subset of B
and B equivalent to a subset of A then A and B are equivalent. This
theorem was also proved by Felix Bernstein and independently by E.
Schroeder."
As you all know, this famous result is almost universally known as
the "Schroeder-Bernstein theorem". But if Cantor did prove it, as
stated above, why his name is not mentioned? I do not remember now
where I saw "Cantor-Schroeder-Bernstein theorem", but I know of no
other text (written before the 1970s) where it is said that Cantor
did really <<prove>> it.
Let me examine with you some authoritative sources with which I am
very well acquainted. According to Abraham Fraenkel in his book
"Abstract Set Theory" (1952), "the first complete proof" of the
theorem was given by Bernstein in 1897 and published in 1898 in
Borel's "Lecons sur la theorie des fonctions". We are also informed
that in 1897 Schroeder gave a "similar" proof which, however, was
"defective" (although Fraenkel is not explicit on this point). On
the other hand, I know from other sources that Cantor had already
guessed the theorem earlier. Thus, to do historical justice, the
theorem would deserve to be called "Cantor-Bernstein theorem". Now,
Cantor in fact tried to prove the result in his final double treatise.
So the next thing to ask is: Is Cantor's argument incomplete in some
respect? To get an answer we will have to examine his proof itself.
Cantor's argument is as follows (according to the well-known
Jourdain's translation):
"We have seen that, of the three relations a = b, a < b, b < a, each
one exclude the two others. (...) Not until later, when we shall have
gained a survey over the ascending sequence of the transfinite
cardinal numbers and an insight into their connexion, will result the
truth of the theorem:
A. If a and b are any two cardinal numbers, then either a = b or
a < b or a > b.
From this theorem the following theorems (...) can be very simply
derived:
B. If two aggregates M and N are such that M is equivalent to a
part N1 of N and N to a part M1 of M, them M and N are equivalent."
Note that B is precisely the result under discussion. But as we can
now see, here Cantor is simply saying that A implies B and that he
will prove A later. However, he never did it! That is why his proof
was considered unsatisfactory. Indeed, theorem A (that any two sets
are comparable) was proved (modulo the axiom of choice) by Zermelo in
1904. On the other hand, Bernstein's proof is "complete" in that it
is entirely independent of the existence of a well-ordering between
arbitrary cardinals. In this sense, therefore, Cantor did not prove
the Schroeder-Bernstein theorem. But even accepting the comparability
hypothesis, his chain of reasoning is not fully cogent.
We can get some more exciting information in Zermelo's papers "A new
proof of the possibility of a well-ordering" and "Investigations in
the foundations of set theory I" both published in 1908 (and
reproduced in Heijenoort's anthology). Referring to the "
Schroeder-Bernstein theorem", Zermelo says that in 1906 he discovered
a proof that "rests solely upon Dedekind's chain theory" and which
"unlike the older proofs by Schroeder and F. Bernstein as well as the
latest proof by Konig (1906), avoids any reference to ordered
sequences of order omega or to the principle of mathematical
induction". He also mentions a "quite similar" proof published by
Peano in that same year. What nobody knew was that they had discovered
essentially a proof that had already been obtained by Dedekind in 1887
and 1899, but which would be published only in 1932. So Bernstein's
proof was not really the "first complete" one. In many disguised
forms, Dedekind's proof was rediscovered and published in many numbers
of the American Mathematical Monthly.
Carlos Cesar de Araujo.
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