Mathematicians often try to classify things. Sometimes this is possible and sometimes it is provably not possible, in a way to be made precise.
On another front, one of Hilbert's problems called for a finite procedure that, given a finite set of generators and relations for a group and a word on the generators, would tell whether the word was equal to the identity element or not. Although it sounds simple enough, it was proved many years ago that there is, in general, no such procedure. However, for some particular groups one is possible. Such groups are called computable groups.
This so-called "word problem" for groups was generalized to other sorts of algebraic objects. The objects for which such a procedure exists are called "computable." These objects are generally much more down-to-earth than others of their kind. So what happens when we try to classify them?No previous knowledge of logic will be supposed.
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