Almost Computable Numbers and Sequences

In sciences, computable numbers and sequences form a important part of study. From ancient Fibonacci numbers to modern quantum series, computable numbers and sequences provide a crucial tool to understand our world. However, scientific measurements are often inexact. We often see deviations from a computational theory being classified as errors. While random errors may occur, there is also the possibility that the phenomena at hand may be just almost computable, rather than computable. Thus, almost computable numbers and sequences may reveal the true structure of nature beyond computation.

There are many conjectures whether a object is computable or not, but there is little understanding whether a incomputable object is a little incomputable or wildly incomputable. As a first step of quantification, almost computable numbers and sequences may be introduced as objects that differ from computable objects by a set of frequency zero. Almost computable numbers and sequences give a first idea of what's a little incomputable, as opposed to wildly incomputable objects. For example, it's interesting to know whether arithmetic truths not provable from a axiomatic system form a small part of all arithmetic truths or a overwhelmingly large part.

Some preliminary results about almost computable numbers and sequences are easily obtained. Almost computable incomputable sequences must differ from a computable sequence in infinitely many places, but the difference can only be of Kolmogorov complexity o(N), where N is the index of the sequence. Almost computable numbers and sequences are uncountable, and form a much richer class than computable numbers. Being a newly proposed class of numbers and sequences, the hierarchy of almost computable numbers and sequences is not yet properly understood. It's a good research problem. For example, it's a good question whether almost computable numbers and sequences are statistically distinguishable from computable counterparts.

Perhaps in some cases, computability is a illusion while almost computability is the reality.