The Löwenheim-Skolem theorem is 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 cardinal
number greater than or equal to (aleph-0). This theorem
established the existence of "nonstandard" models of arithmetic.
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(aleph-0). This theorem
established the existence of "nonstandard" models of arithmetic.
(aleph-0) domain of interpretation.
Hence, aleph-0 domains are sufficient for interpretation
of first-order logic.
