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 Proof by Richard Hammack. ed 3.2
The 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)
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 Proof by Richard Hammack. ed 3.2
The 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)