Cantor has never attempted to formalize set theory axiomatically. Only toward the end of his active career, under the influence of Hilbert, he has considered axioms for set theory at all. "He sees, already toward the end of the 19th century, the appearance of formalistic thinking that he abhorred" [Meschkowski]. The following paragraphs will cover every mentioning, that I am aware of, of axioms in Cantor's work and correspondence.
The so-called Cantor's axiom (1872) concerns geometry only:
... to add an axiom, requiring that, vice versa, to every numerical value there belongs a certain point of the straight the co-ordinate of which is equal to that numerical value. ... I call this sentence an axiom because it is immanent to its nature that it cannot be proven in general. [G. Cantor: "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen", Math. Annalen 5 (1872) 123 - 132]
Cantor considered the Archimedian axiom a provable theorem:
Thus the "Archimedean axiom" is not an axiom but a theorem that follows with logical necessity from the notion of linear magnitude. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer (1932) p. 409]
I maintain the position that the so-called "Archimedean axiom" has been proven by myself and that a deviation from this "axiom" means going astray. [Cantor to Veronese, 7. Sep. 1890]