( 787 ) 
it with the aid of the notion of the second tramjinite cardinal St, which 
is however not recognised by all mathematicians. Lindelof ') gave a 
proof independent of this notion, where, however, the process of 
destruction itself remaining non-considered, the result is more or less 
obtained by surprise. 
Only for linear sets there have been given proofs of the fundamental 
theorem, which at the same time follow the process of destruction 
and are independent of St z ). 
The rest set which remains after completion of the process of 
destruction and which we may call the Cantor residue, is after 
Cantor *) a perfect set of points, however of the most general kind, 
thus in general not a perfect set of pieces. 
An extension of the fundamental theorem, enunciated by Schoenfues * * * 4 5 ) 
and proved by me ■), can be formulated as follows: 
If we destroy in a closed set an isolated piece, in the rest set again 
an isolated piece, and so on transfinitely, this process leads after a 
denumerable number of steps to an end. 
My proof given formerly for this theorem was a generalisation of 
Lindelof’s method, but at the same rime I announced a proof 
which follows the process of destruction, and which I give now here; 
in it is contained a proof of the fundamental theorem, which in 
simplicity surpasses by far the existing ones, is independent of &, and 
follows the process of destruction: 
By means of Sp n - i’s belonging to an orthogonal system of directions 
we divide the Sp n into w-dimensional cubes with edge a, each of 
these cubes into 2" cubes with edge - a, each of the latter into 2" 
cubes with edge — a, etc. 
All cubes constructed in this way form together a denumerable 
set of cubes K. 
Let now p be the given closed set, then K possesses as a part a 
likewise denumerable set K t consisting of those cubes which contain 
in their interior or on their boundary points of p. 
>) ScHoranujES^Bericht flber die Mengenlehre I, f *?■ 8 * i G6lt ' Na ^^ 1903 ’ 
21—31; Hardy, Mess, of Mathematics 33, p. 67—69; Young, Proceedings of 
the London Math. Soc. (2) 1, p- 230-246. 
») 1. c. p. 465. , . 
' 4) Mathem. Annalen 59; the proof given there 
die Mengenlehre H, p. 191-1* does not hold. 
5) Mathem. Annalen 68, p. 429. 
p. 141-145, and Bericht uber 
