Atheism is most definitively and explicitly a belief. It doesn't have any associated hierarchy and not much in the way of dogma, so may you can argue that it's not a belief system, but it is by its very definition a belief.
That implies that the number of beliefs I hold is infinite. Furthermore, I only knowingly hold some of those beliefs and the fraction of total beliefs that I hold knowingly is, essentially, zero.
Only in a trivial sense that certain sets contain an infinite number of members. But you have had a finite number of thoughts, and if you don't think about something, you don't have a belief about it.
I don't even know what a believer is thinking about when they think about a god (only they do) so I am not thinking about it therefore I am not an atheist?
What I am addressing in part, as I am sure you must realize, is burden of proof in the manner of
Russell's celestial teapot argument. You seem to be suggesting (and I certainly might be misunderstanding you) that rejecting belief in something implies thinking about that something. I contend that belief in god(s) is like belief in woodland fairies (or celestial teapots) and that rejecting belief in god(s) is like rejecting belief in woodland fairies (or celestial teapots). But it doesn't have to be a teapot or a woodland fairy, it can be anything for which we have no good reason to believe in its existence. In effect, for the category of things which lacks any supporting evidence, even if we often cannot have true logical disbelief in them (because we really cannot logically rule out a celestial teapot) we can have something that is very much like it as a consequence of where the burden of proof lies and thus reject the belief.
But the point of
Russell's celestial teapot is not to show that celestial teapots do not exist (or, more properly, that they are so unlikely to exist as to make perverse the assignment of any significance to the possibility of their existence) but to show, by analogy, that belief in other things which are like a celestial teapot (for instance, a celestial tea cup) is equally unwarranted. My point though, is that this applies even if you are not specifically thinking of a particular kind of tableware. Therefore, if I have not ever specifically thought about it, it does not mean that I am agnostic about its existence until you bring it up and I think about it. Under this circumstance, I have justification to
a priori reject the existence of these things and therefore I will not call myself agnostic on this issue.
In other words, disbelieving in celestial teapots also implies disbelief in other kinds of tableware even if I am not thinking about celestial tableware at all. That is, not believing in something is not necessarily a type of thought about that something (just like not playing chess is not a kind of game). The beliefs I do not hold are infinite (just like there are an infinite number of games one could play which haven't even been created) but I am not thinking about an infinite number of things (just like no one plays an infinite number of games --as would be the case if not playing a game was in itself a type of game).
Let's put it differently. I am an atheist. I lack a belief in Zeus. My contention is that I lack a belief in Zeus because there are no compelling reasons to believe in such a being. Your contention seems to be that it is
only because I can think about Zeus so that I may actively disbelieve him that I can call myself an atheist. But I contend that (unless you can give me compelling reasons to do otherwise) I am also rejecting a Zeus like god who shoots cobalt blue colored lightning bolts (rather than regular lightning bolts), or a Zeus like god who shoots bolts of lightning only in pairs or any of an infinite number of Zeus like gods which I have not ever even thought about.
I can most definitely lack a belief in something which I call gods without thinking about it. If I could not do that, I could never be an atheist because there are always possible gods which I have never thought about (for which we also happen to lack any reason to think they exist) which I would have to actively disbelieve in order to truly be able to call myself an atheist.
If the answer is yes, then you're an atheist. If the answer is no, then you're not an atheist. Quite simple, really. And it hinges upon your belief, which you haven't told me. So I can't answer your question.
No, it hinges upon a lack of belief.