So if I count the negative integer set and then count beyond them, would that 'next' number be negative omega (-ω)? There would be no negative aleph-null though right? Like the size of the set of negative integers is still positive aleph-null?
Is any literature published on negative transfinite numbers or do they not exist at all?
Thanks so much!
Ordinals are really just order types of specific orders. In this context we usually denote $\alpha^*$ as the reverse order type of $\alpha$. So $\omega^*$ is the order type of the negative integers.
Yes, the concept is sound, but the question is, what do we want to do with it.
Your use of "negative" suggests somehow that this should be incorporated into ordinal arithmetic, which again is a smaller part of "order arithmetic", but this is certainly not what you would expect from normal addition. For example $\omega+\omega^*$ is not $0$, but rather the order type of having the positive integers and after them the negative integers, concretely this would be $\{\frac1k\mid k\in\Bbb Z\setminus\{0\}\}$ as a set of real numbers.
To add insult to injury, even if you want to think about negative ordinals as some formal objects used for subtracting one ordinal from another, then you need to talk about the pair you're subtracting, not just the one ordinal. For example, $\omega-1$ would be the order type of $\omega\setminus 1$ which is again $\omega$, and so $\omega+1-1$ is also $\omega+1$, since we subtract an initial segment and not an end segment (otherwise $\omega-1$ is not defined).
And this is why the comments mention the surreal numbers. These numbers form a field, and the ordinals embed into them (and they play the same role that the integers play in the real numbers as some sort of "backbone"). But now since the surreal numbers form a field, every ordinal has an inverse (both additive and multiplicative (with the exception of $0$ on the latter)), and these would behave slightly closer to what you would expect. But one should note that the ordinal arithmetic and the surreal numbers are generally incompatible.