Computable and Definable Sequences
An integer sequence is a computable sequence, if there exists an algorithm which given n, calculates an, for all n > 0. An integer sequence is a definable sequence, if there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences. The set of computable integer sequences and definable integer sequences are both countable, with the computable sequences a proper subset of the definable sequences (in other words, some sequences are definable but not computable). The set of all integer sequences is uncountable (with cardinality equal to that of the continuum); thus, almost all integer sequences are uncomputable and cannot be defined.
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