For a homework I have been ask to prove that if a theory $\Sigma$ over a vocabulary $L$ has a model with countable domain, then $\Sigma$ has a model with uncountable domain.
I have no idea how to proceed, any suggestions are welcome.
Thanks.
2026-04-13 14:03:39.1776089019
If a theory over a vocabulary $L$ has a model with countable domain, then it has a model with uncountable domain
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Hint: Add to the theory uncountably many new constant letters, plus axioms stating that they are pairwise different. Now use compactness.