"Mächtigkeit" versus "Kardinalität"? In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
 A: Here is Cantor's Beiträge zur Begründung der transfiniten Mengenlehre (Erster Artikel). Read the bottom four lines on the first page: ",Mächtigkeit' oder ,Cardinalzahl' von $M$ nennen wir $\ldots$". That looks a lot like Cantor intended the two to be names for the same concept.
The two lines above that reinforce that idea.
A: The two terms "Mächtigkeit" and "Kardinalität" do indeed mean the same in Cantor's set theory; also today, for instance in lectures in German and Swiss universities, the terms are used as synonyms for the same concept.
A: Here is how Cantor introduced "Mächtigkeiten" in Ueber eine elementare Frage der Mannigfaltigketislehre (1890):

The "Mächtigkeiten" represent the unique and necessary generalisation of the finite "Cardinal numbers", they are nothing other than infinitely large Cardinal numbers, and they share the same reality and definiteness.
So it seems that, at least initially, Cantor did not speak of "Kardinalität", which was linked to the existing term of "cardinal numbers", a term from the 16th century meaning "principal numbers". He introduced a new term "Mächtigkeiten" for infinitely large cardinal numbers. The dictionary I consulted lists 1935 as the first use of the term "cardinality". Because "Mächtigkeiten" is not easily transferred to the English language, the shift to "Kardinalität" and "cardinality" seems a natural one.
