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Post Closed as "Needs details or clarity" by Andrés E. Caicedo, Ricardo Andrade, Joseph Van Name, Ryan Budney, Andrey Rekalo
Edit: changed countable with sigma-complete (including in title)
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Countable Sigma-complete Lindenbaum algebras?

Is there any calculus whose algebraization is a countablesigma-complete Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?

Countable Lindenbaum algebras?

Is there any calculus whose algebraization is a countable Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?

Sigma-complete Lindenbaum algebras?

Is there any calculus whose algebraization is a sigma-complete Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?

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Countable Lindenbaum algebras?

Is there any calculus whose algebraization is a countable Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?