Comment by dwohnitmok

My other reply is so long that HN collapsed it, but addresses your particular question about how to create the mapping between finite-length strings and the real numbers.

Here's another lens that doesn't answer that question, but offers another intuition of why "the fact that there are a countable number of strings and an uncountable number of reals" doesn't help.

For convenience I'm going to distinguish between "collections" which are informal groups of elements and "sets" which are formal mathematical objects in some kind of formal foundational set theory (which we'll assume for simplicity is ZFC, but we could use others).

My argument demonstrates that the "definable real numbers" is not a definition of a set. A corollary of this is that the subcollection of finite strings that form the definitions of unique real numbers is not necessarily an actual subset of the finite strings.

Your appeal that such definitions are themselves clearly finite strings is only enough to demonstrate that they are a subcollection, not a subset. You can only demonstrate that they are a subset if you could demonstrate that the definable real numbers form a subset of the real numbers which as I prove you cannot.

Then any cardinality arguments fail, because cardinality only applies to sets, not collections (which ZFC can't even talk about).

After all, strictly speaking, an uncountable set does not mean that such a set is necessarily "larger" than a countable set. All it means is that our formal system prevents us from counting its members.

There are subcollections of the set of finite strings that cannot be counted by any Turing Machine (non-computably enumerable sets). It's not so crazy that there might be subcollections of the set of finite strings that cannot be counted by ZFC. And then there's no way of comparing the cardinality of such a subcollection with the reals.

Another way of putting it is this: you can diagonalize your way out of any purported injection between the reals and the natural numbers. I can just the same diagonalize my way out of any purported injection between the collection of definable real numbers and the natural numbers. Give me such an enumeration of the definable real numbers. I change every digit diagonally. This uniquely defines a new real number not in your enumeration.

Perhaps even more shockingly, I can diagonalize my way out of any purported injection from the collection of finite strings uniquely identifying real numbers to the set of all natural numbers. You purport to give me such an enumeration. I add a new string that says "create the real number such that the nth digit is different from the real number of the nth definition string." Hence such a collection is an uncountable subcollection of a countable set.