On the history and restrictions of the diagonal method

This work describes the history of occurrence of the diagonal method of reasoning about absolutely ordered sequences. The occurrence of such reasonings corresponds to the 5.3 subperiod of the scientific methodology development. It is shown that diagonal reasonings proved only the impossibility of the everywhere-dense linearly ordered list of a countable set, which is illustrated by the example of realization of the separation axiom (Hausdorff axiom) on a countable set of decimal notations. Similar restrictions of the diagonal method also take place when applying it to reasonings in formalized theories (when using Godel's numbering, etc.).