Integer sequence

An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates a_n, for all n > 0. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.

Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.

Suppose the set M is a transitive model of ZFC set theory. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences. In each such M, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.

For some transitive models M of ZFC, every sequence of integers in M is definable relative to M; for others, only some integer sequences are (Hamkins et al. 2013). There is no systematic way to define in M itself the set of sequences definable relative to M and that set may not even exist in some such M. Similarly, the map from the set of formulas that define integer sequences in M to the integer sequences they define is not definable in M and may not exist in M. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model (Hamkins et al. 2013).

If M contains all integer sequences, then the set of integer sequences definable in M will exist in M and be countable and countable in M.

A sequence of positive integers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

Integer sequences that have their own name include:

• On-Line Encyclopedia of Integer Sequences
  • List of OEIS sequences

• Hamkins, Joel David; Linetsky, David; Reitz, Jonas (2013), "Pointwise Definable Models of Set Theory", Journal of Symbolic Logic, 78 (1): 139–156, arXiv:1105.4597, doi:10.2178/jsl.7801090, S2CID 43689192.

• Journal of Integer Sequences. Articles are freely available online.