Re: Constructibility of X -> X^2 bijection



Daryl McCullough wrote:

Stephen J. Herschkorn says...



Do you consider the prood of Cantor-Shroeder-Bernstein to be constructive? If so, Kunen shows how to build an injection from k x k to k when k is an infinite cardinal



That's the part that I was wondering about. What is the injection
from k x k -> k?

Of course, I know of such injections for the case
k = omega and k = 2^omega, but those don't obviously
generalize to other cardinals.


Ok, here it is, from p. 29 of Kunen: By definition, a cardinal is an ordinal. We define a relation R on k x k:
((a,b), (c,d)) in R iff
[maj(a,b) < maj(c,d)] or [maj(a,b) = maj(c,d) and (a,b) precedes (c,d) lexicographically]
Show by induction that R is a well-ordering of type no greater than k.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

.

Quantcast