Abstract
We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake, we prove that the theory of densely ordered sets with generic predicates is \(\aleph _0\)-categorical. A model of the theory and a particular continuous choice-function is constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a by-product, other functions with surprising properties can be constructed.
Highlights
All sciences, including mathematics and computer science, and most of their practical applications, are supported by numeric computations
The numbers expressible with a finite amount of digits are rational numbers, and so they can only approximate the exact values of continuously varying quantities
That is why we choose to focus on exactly represented rational numbers only
Summary
All sciences, including mathematics and computer science, and most of their practical applications, are supported by numeric computations. 6. Proof: Q+ is countable and is a model of the theory T of dense ordered sets without endpoints. A particular countable dense ordering with generic predicate is constructed, together with a continuous choice-function.
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