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It is a misconception to think that the list is finite because it is indexed by a natural number. The natural numbers are “closed under addition”. It means that you can (for example) add 1 indefinitely, and you still have a natural number.

Each block in the enumeration gives an extra digit. The list is not finite, and so the number of digits is also not finite.

According to a standard text:

Theorem 14.3:A set is countably infinite if and only if its elements can be arranged in an infinite list a1, a2, a3, a4, …Book of Proofby Richard Hammack. ed 3.2The point is that this method of bijection is not a valid counting method.

See:

The Misuse of Bijection when Comparing Infinite Sets, (2022), Green. L.O. (https://www.researchgate.net/publication/360504965)