# [FOM] Paul Cohen was wrong

William Tait williamtait at mac.com
Tue Sep 13 10:48:43 EDT 2011

On Sep 12, 2011, at 11:11 AM, Monroe Eskew wrote:
> Without power sets, what mathematical reason would we have for
> asserting that different sizes of infinity exist at all?  Cantor
> established their existence using the assumption that the set of all
> real numbers exists.

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