There is an interesting comment by Gauss on "infinite magnitude as a complete thing" that has invited varying interpretations. In a well-known passage, Gauss criticized the use of infinity in mathematics in the following terms:
"I protest first of all against the use of an infinite quantity as a completed one, which is never permissible in mathematics. The infinite is only a façon de parler, where one is really speaking of limits to which certain ratios come as close as one likes while others are allowed to grow without restriction."
This has often been interpreted by historians such as Dauben as rejection of "completed" infinity, also known as actual infinity (and contrasted with potential infinity). Actual infinity is considered to have been "tamed" by Cantor, who based his theory of infinite ordinals and cardinals on the idea of actual infinity. Hilbert famously referred to such a framework as a "Cantorian paradise".
However, other historians give a different interpretation of Gauss' passage. Waterhouse points out in
WILLIAM C. WATERHOUSE, GAUSS ON INFINITY, Historia Mathematica 6 (1979), 430-436
that the context of Gauss's passage is a criticism of a contemporary mathematician's (Schumacher) attempted proof of the parallel postulate, using the idea of a circle of a radius of infinite magnitude. Accordingly, it is not infinite cardinals but rather infinite magnitudes that are rejected by Gauss. Such infinite magnitudes were formalized in nonstandard analysis, and can be called infinite ringinals (because they are elements of a ring such as the ring of integers), to contrast them with Cantorian infinities.
Meanwhile, user Conifold (who seems to be a professional historian) argues vigorously that Waterhouse's argument is inconclusive, and Dauben (and many other historians) may well be right.
This leads us to the following questions:
1. Did Gauss criticize infinite cardinals or infinite ringinals?
2. Are there any texts by Gauss that could shed light on his attitude toward what Leibniz termed "infinite wholes" (as opposed to infinite magnitudes)?
Moore notes the following:
"Waterhouse [1979] has noted in this journal, correctly, that Gauss’s remarks were not directed against set theory, which in any case did not exist at that time. But Cantor saw Gauss’s remarks as aimed against the actual infinite, etc."
See Moore, Gregory H.$\;$ Hilbert on the infinite: the role of set theory in the evolution of Hilbert's thought. Historia Math. 29 (2002), no. 1, 40–64.
In more detail, Lipschitz understood Cantor as interpreting the Gaussian passage as opposition to actual infinity (i.e., completed infinite collections). Waterhouse argues that Lipschitz's comment is the source of the commonly found opinion that Gauss opposed actual infinity (as opposed to potential infinity; both of these are Aristotelian notions). The question is whether in the quoted passage, Gauss opposed (A) actual infinite collections or (B) infinite magnitudes. In modern mathematics, these are formalized respectively by (A) cardinals and (B) nonstandard numbers (which can be referred to as "infinite ringinals" because they are elements of a ring).
It should be noted that the existence of infinite cardinals is, mathematically speaking, "transverse" to the existence of "infinite ringinals"; see e.g., Enayat's answer. Historically, Leibniz used infinite magnitudes (that he referred to as infinita terminata) but rejected completed infinities that he referred to as "infinite wholes", already in the 17th century. In the 18th century, Euler made routine use of infinite quantities.
While there is no evidence that Gauss distinguished between the notions of "infinite magnitudes" and "infinite sets" in his letter to Schumacher, it is legitimate to ask what notion of infinity Gauss was referring to in this letter, based on the context of the Gauss-Schumacher discussion.
I posed the question at the philosophy SE where it did not generate any answers, and at the history of science SE where it generated one answer which seems to lack mathematical expertise to distinguish between different types of infinity.
Waterhouse's paper on Gauss was reported on here.
