Set Theory and Logic

A Paradox of “Adjacent” Real Points and Beyond

Authors: Ke Zhang
Comments: 5 pages. Email: alspa@163.com

We reveal adjacent real points in the real set using a concise logical reference. This raises a paradox while the real set is believed as existing and complete. However, we prove each element in a totally ordered set has adjacent element(s); there is no densely ordered set. Furthermore, since the natural numbers can also be densely ordered under certain ordering, the set of natural numbers, which is involved with each infinite set in ZFC set theory, does not exist itself.
Category: Set Theory and Logic