Re: Non-countability of R

In article
<716f834e-67f8-4259-b5b3-3b3c1d3bea3a@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote:

William Elliot wrote:

There's a bijection between those and P(N).
Seems likely that the theorem you reference came
before his reknown |S| < |P(S)|.

Virgil wrote:

I believe so, but do not have dates.

From a 13 May 2007 sci.math.research post of mine:

It seems that the first person to make an explicit
connection between Cantor's 1892 diagonalization
result and the collection of all subsets of a set
was Bertrand Russell in 1903.

I learned this from José Ferreirós's 1999 book
"Labyrinth of Thought: A History of Set Theory
and its Role in Modern Mathematics" (top of p. 306),

who writes:

"It has to be noted that it was Russell, not Cantor in his
published work, who focused on the Cantor Theorem as a
central result of great importance. He seems to have been
the first mathematician who presented it as showing that
the set of all subsets of S has always a greater cardinality
than S itself [Russell (1903), Sections 346-347]. Thus,
it was Russell who formulated it for the first time as a
purely set-theoretical result. (In Cantor's version it
showed that, given a set S, a certain set of _functions_
has greater cardinality, and functions were _not_ taken
to be sets.) In the process, Russell was the first to
emphasize something like the Power Set Axiom. All of
this was in itself an important contribution, for until
then the theorem lay rather forgotten in the first annual
report of the DMV and its significance had not been
clearly grasped."

However, I think Ferreirós may have overlooked Borel's
1898 book in this regard. After getting a copy of this
book (from a local university library) to more carefully
look over the part about Cantor's diagonal proof,
which I had previously seen and thus knew was in Borel's
book, I see that Borel also calls attention to the
connection between Cantor's 1892 proof and the collection
of all subsets of a set. In the last section of his
"Note I: La Notion Des Puissances" (pp. 102-110),
titled "La puissance des ensembles de fonctions"
(pp. 109-110), Borel clearly introduces the notion
of a characteristic function (but doesn't name the
notion -- this was due to Vallee-Poussin in 1915,
I think, who also made strong use of the idea in
measure theory) and shows how Cantor's proof implies
that the collection of all subsets of the reals has
cardinality greater than the set of reals.

Dave L. Renfro

Thanks.
.

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