# [FOM] Paul Cohen was wrong

Monroe Eskew meskew at math.uci.edu
Tue Sep 13 13:09:31 EDT 2011

But Cantor proved that the reals are uncountable much earlier, in the
1874 paper "Über eine Eigenschaft des Inbegriffes aller reellen
algebraischen Zahlen" ("On a Property of the Collection of All Real
Algebraic Numbers").

Monroe

On Tue, Sep 13, 2011 at 7:48 AM, William Tait <williamtait at mac.com> wrote:
>
> In his construction of the alephs in 1883, Cantor did not use power set. He defined kappa^+ as the LUB of the least set X such that
> 1. 0 \in X
> 2. If alpha \in X, then alpha+1\in X
> 3. If beta < or = kappa and f : beta ---> X, then LUB{ f(alpha) | alpha < beta} \in X.
>
> Later, in his 1890-91 paper on the diagonal method, he constructed the beths
>  using power set (in the form of assuming that for each set Y the set of 2-valued functions on Y exists).
>
> Bill Tait
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