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”.
This is a rabbit hole with no end.
This is a rabbit hole with no end.
This does not hold if you demand that, for example, the map
k -> a_k
that represents the Cauchy sequence, is a computable function.
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.
Edit: oops, not that. The sequence itself - not the "useful reals" element of the sequence - has to be non-constructive...
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".
Exploring the consequences of alternative definitions is fantastic, let’s do more of that.
Redefining the terms that somebody else uses in a conversation sucks royally. Everybody hates it when people do that. Don’t be that guy.
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".
Does that seem OK to you?