If I have a sequence $a_0, a_1, a_2, \cdots$ , then is expressing the limit of this sequence as $a_\omega$ sensible?

Question

If I have a sequence created by some rule which comes to a limit , then I can express it as $a_0, a_1,a_2,\cdots$.

If I said $\lim_{n \to \infty} a_n = a_{\omega} $ , is that a sensible thing to do ?
Would extending that into $a_\omega, a_{\omega+1},\cdots$ make sense , especially if every time I took a limit , I would go up a limit ordinal?

Answer 1

You have hit upon one of the fundamental ideas of infinitesimal analysis. However, it needs to be explained neither in terms of infinite ordinals nor in terms of cardinals, but rather in terms of infinite numbers in the ring of integers.

Thus, instead of thinking of $\omega$ as an ordinal, think of it as an infinite (more precisely, unlimited) number in the ring ${}^\ast \mathbb Z$ of hyperintegers. The basic point is that the limit of a sequence $(s_n)$ is the standard part of $s_\omega$. Thus the limit will not in general be equal to $s_\omega$ but at any rate will be infinitely close to $s_\omega$. Here $\omega$ needs to be chosen as an unlimited integer but the answer (i.e., the standard part) is independent of the choice of $\omega$.

This works for all sequences, including the sequence of truncations of the decimal expansion of $\pi$ mentioned in the comments. Namely, by the extension principle, the decimal digits are also defined at unlimited ranks such as $\omega$ chosen above.

Answer 2

(1) When you are considering $n \to \infty$ to indicate the normal usage [ like arbitrarily large Integer , arbitrarily large real number , extended real number ] , then it might be more sensible to use the notation $a_\infty$

(2) When you are considering $n \to \infty$ to indicate the larger infinity values [ like the trans-finites & non-standard analysis & variations ] , then it might be more sensible to use the notation $n \to \omega$ , because $\infty$ is ambiguous.
When you use that notation , then it is sensible to say $a_\omega$ & we can consider $a_{\omega+1}$ , $a_{\omega+2}$ ,$a_{2\omega}$ ,$a_{2\omega+2}$

(3) We should not mix up these 2 Cases to keep things Consistent & meaningful.

(4) Of course , you should ensure that $n \to \infty$ & $a_\infty$ versus $n \to \omega$ & $a_\omega$ make sense within your Scenario.

Noah Schweber gave a great Example , that there are "Countable" Digits for $\pi$
Likewise , we can not generally consider $\sin \omega$
While $e^{-\infty} \equiv 0$ , it is not easy to see whether we want $e^{-\omega} \equiv 0$ too.

Qiaochu Yuan has mentioned a great Point , that you might have to give Explanations.
You have to give the Definitions , making sure all the infinite & infinitesimal values are consistent & meaningful , within your Scenario.