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A fundamental result in Model Theory which states that if a countable theory has a model, then it has a countable model.
Furthermore, it has a model of every Cardinality greater than or equal to 
(Aleph-0). This theorem
established the existence of ``nonstandard'' models of arithmetic.
See also Aleph-0, Cardinality, Model Theory
References
Chang, C. C. and Keisler, H. J. Model Theory, 3rd enl. ed. New York: Elsevier, 1990.