There are uncountably many Cauchy sequences of useful reals. You can’t write them all down. So now you have to also restrict yourself to “useful Cauchy sequences of useful reals”.
Set theorists have located the "end" for all practical and most impractical purposes. Let M be the minimal countable transitive model of ZFC. Declare a real to be useful if and only if it is in M.
I am a bit shocked, because that seems like a really dishonest way of doing things. “Surprise! Actually, I am not talking about Cauchy sequences, but only computable Cauchy sequences.”
If you change the rules you had better be up front about it.
What you are describing is a completely different definition for “complete metric space” than what is commonly accepted by the mathematical community at large. So do not be surprised that by using different definitions, you come to different conclusions.
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I am a bit shocked, because that seems like a really dishonest way of doing things. “Surprise! Actually, I am not talking about Cauchy sequences, but only computable Cauchy sequences.”
Rather: If countability is important to you, you should change the rules so that the property that the field is closed w.r.t limits of Cauchy sequences does not make your set uncountable.
Redefining the rules if something does not work is how you do mathematics works all the time:
- A PDE does not have a solution in a classical sense and you hate this? No problem: You invent the theory of weak solutions and distributions and simply change the concept what is to be considered a solution of the PDE.
- The concept of algebraic varieties turns out to be to limiting to obey the rules that you would love them to have? No problem: You define the concept of algebraic schemes and now talk about algebraic schemes instead of varieties (https://en.wikipedia.org/w/index.php?title=Scheme_(mathemati...).
TLDR: Mathematics is often the art of "defining your problems away".
> If countability is important to you, you should change the rules so that the property that the field is closed w.r.t limits of Cauchy sequences does not make your set uncountable.
Ok, fine: I hereby declare that the set Q of rationals is a closed field, because I define "closed" to mean "closed under Cauchy sequences whose limit points are rational numbers".
This is a rabbit hole with no end.