Wayne Martin was kind enough to post a very interesting (and fantastically clear) paper on Hegel's notion of "bad" infinity versus a "true" infinity (sometimes rendered as "spurious" versus "genuine" infinity). Or in German schlect Unendlichkeit versus wahrhaft Unendlichkeit. His analysis centers mainly on the larger Logic and its section on infinity which is part of chapter 2 "Determinate Being," section c. (a) through (c). Anyone wanting a refresher or just a clear articulation of Hegel's at times just incredibly cumbersome discussion of the metaphysics of infinity ought to read the paper, which interestingly has some larger goals in mind (such as providing a Fichtean response to Hegel's pre Phenomenology of Spirit essay "The Difference Between Fichte's and Schelling's Systems of Philosophy" of 1801).
In a nutshell a bad infinity, for Hegel, is just one that is open-ended. For Hegel true infinity - accessible by reason, but still beyond the understanding - is a totality. Intensive and extensive mathematical infinities, such as those infinitesimally available between any two numbers or those which are sets of numbers to be aggregated indefinitely to any other set, fail before the sort of actual infinity that is an absolute total requiring nothing outside of itself.
The interesting part isn't Hegel's critique of the bad infinite as one that is a possible infinite. It is that Hegel states an infinity that "sets itself over and against" any other (such as the finite) is bad precisely because it lacks an infinite nature that is its own and is therefore delimited by negating something else so as to take on its own identity. So the logic of contrasts that establishes an infinite precisely because it is not finite, or "in-finite" is one that necessarily fails having no essential nature that is, itself, properly infinite or total.
Hegel compares the "true" infinite for this reason to a circle. A circle does not "go on and on forever" but rather is "unending" in the sense that it is a closed actual total. This total encompasses the finite, or contains it, but is in no way limited by it. God, for Hegel, in traditional definitions, fails to capture the logical meaning of true infinity because God's nature and infinitude is always set over and against finite creation and thus is always dependent upon it to be what it is. God in that sense is limited and finite, but not truly infinite.
A final thought. As the paper's title is "In Defense of Bad Infinity" it seemed appropriate to me to recommend in contradistinction to Hegel's infinite Schelling's take on infinity, for it was Schelling who was the first German idealist I had encountered even before Hegel. For a time in fact it was Schelling's "bad infinity" that served as a model for me to attempt to understand Hegel's own division between true infinite and spurious infinite.
To whit, then: might we be able to define the absolute not necessarily as something unending such as in an unending series of numbers (a potential infinite) or on the other hand as something total as in a closed circle (an actual infinite). Perhaps, rather, might not the absolute be something absolutely unconditioned as Schelling ventured to maintain? That is, an unconditioned infinite? What might this mean?
It was Schelling's view that absolute being - "the Absolute" - must mean unconditioned being, and as unconditioned being, the absolute, must therefore be "endless" for any "beginning" or "end" of it would qualify some condition as being other to that absolute's identity and the result would be a non-absolute identity. Here we see that for Hegel the absolute contains its own identity as well as all else, and this is what makes it "genuine" or "true" (wahrhaft Unendlichkeit). In this way we might say its infinite nature is of a pure metaphysical positivity of sorts. Schelling's absolute, on the other hand, is said to be "spurious" for it is absolutely indifferent to any particular identity in the sense that nothing can be other to it save only for itself. Paradoxically to be "itself" it must both be "itself" and "not itself" in different respects (so that there is no "other" conditioning it). Therefore, Schelling's absolute is "endless" but without closure as a totality because there is nothing other that it could be other than its own identity, but also quite strangely, its "non-identity"; that is, it accommodates for "that which it is not." In this way we might say that Schelling's absolute is "in-different." And so with that said, why might Schelling's notion of the absolute be preferable to Hegel's?
Hegel's absolute is a thing, it is an identity: the Absolute, the All, which contains all things. But Schelling's main point was that the absolute, unlike in Hegel's philosophy, could never be a totality or a thing. Schelling claimed instead that the absolute was no-thing at all! Here Hegel railed against Schelling for thinking that the absolute had no identity sensu stricto and raved that as being "nothing" Schelling's absolute could only be a blanket that simply absorbed all things in its cloak of night, in a "blank" identity, where those particular things would not be preserved within the circle of infinity. This is to say that Hegel was fearful that Schelling's absolute annulled particular things within an indifferent identity instead of raising them up and encircling things within the identity of an absolute that is itself complete and total. Thus Hegel claimed that Schelling's absolute was an identity that was a "night in which all cows are black."
In short, then, it seems that Schelling's infinite or absolute is conditioned by no other indeed. But... it is "absolutely unconditioned" as indifferent to identity in its being unconditioned. And this, therefore, can be said be its positive. Schelling's unconditioned infinite is absolute indifference; that is, it is unconditioned ground in perpetuity and as such it nevertheless exhibits a sort of metaphysical positivity that Hegel claimed it was lacking.
Should we agree with Hegel that this lack of identity within Schelling's notion of the absolute means that his concept of the absolute as infinite is deficient as compared to Hegel's own understanding of the absolute? Or, might it be that in this very deficiency, in this "spurious" nature lies hidden a more adequate "positive" conception of the absolute and hence its virtue?
Paper "In Defense of Bad Infinity" HERE.