Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice
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Dzik  gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems, here called Dzik systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the conditional \( let \Rightarrow LET \). A partial solution is provided.
Keywords.2-compactness axiom of choice consequence operation deductive theory Dzik system equilateral consequence operation extremal consequence operation Lindenbaum punctiliousness saturated theory
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