This about the famous article

Zermelo, E., *Beweis, daß jede Menge wohlgeordnet werden kann*, Math. Ann. 59 (4), 514–516 (1904),

available here. **Edit:** Springer link to the original (OCR'ed, may be behind a paywall)

An English translation can be found in the book **Collected Works/Gesammelte Werke** by Ernst Zermelo. Alternative source: the book **From Frege to Gödel: a source book in mathematical logic, 1879-1931**, by Jean Van Heijenoort.

[See also this interesting text by Dan Grayson.]

I don't understand the paragraph on the last page whose English translation is

Accordingly, to every covering $\gamma$ there corresponds a definite well-ordering of the set $M$, even if the well-orderings that correspond to two distinct coverings are not always themselves distinct. There must at any rate exist

at least onesuch well-ordering, and every set for which the totality of subsets, and so on, is meaningful may be regarded as well-ordered and its cardinality as an "aleph". It therefore follows that, for every transfinite cardinality, $$\mathfrak m=2\mathfrak m=\aleph_0\,\mathfrak m=\mathfrak m^2,\mbox{and so forth;}$$ and any two sets are "comparable"; that is, one of them can always be mapped one-to-one onto the other or one of its parts.

It seems to me Zermelo says that the fact that any set can be well-ordered *immediately* implies that any infinite cardinal equals its square. Is this interpretation correct? If it is, what is the argument?

Side question: What is, in a nutshell, the history of the statement that any infinite cardinal equals its square? Where was it stated for the first time? Where was it proved for the first time? (In a similar vein: where was the comparability of any two cardinal numbers proved for the first time?)

**Edit:** German original of the passage in question:

Somit entspricht jeder Belegung $\gamma$ eine ganz bestimmte Wohlordnung der Menge $M$, wenn auch nicht zwei verschiedenen Belegungen immer verschiedene. Jedenfalls muß es

mindestens einesolche Wohlordnung geben, und jede Menge, für welche die Gesamtheit der Teilmengen usw. einen Sinn hat, darf als eine wohlgeordnete, ihre Mächtigkeit als ein „Alef“ betrachtet werden. So folgt also für jede transfinite Mächtigkeit $$\mathfrak m=2\mathfrak m=\aleph_0\,\mathfrak m=\mathfrak m^2\text{ usw.,}$$ und je zwei Mengen sind miteinander „vergleichbar“, d. h. es ist immer die eine ein-eindeutig abbildbar auf die andere oder einen ihrer Teile.