Some definitions of the ``smallest infinity'' (Q2735850)

In this paper the author investigates various definitions for the notion of the smallest infinity. He considers the relationship between these definitions. Most of these definitions for a set \(I\) use linear orderings on \(I\), mappings on \(I\) and chains on \({\mathcal P}(I)\).NEWLINENEWLINENEWLINEThe results for chains seem to be not correct. So property TI2 for \(I\) demands that \(I\) be infinite and every \(\subset\)-chain in \({\mathcal P}(I)\) be equipotent to \(I\). The author states that \(I\) being equipotent to \(\omega\) implies TI2. This contradicts the fact that it is easy to find, in ZFC, a chain in \({\mathcal P}(\omega)\) which has the order type of the reals.