Some restricted lindenbaum theorems equivalent to the axiom of choice

Logica Universalis 1 (1):183-199 (2007)
Abstract
.  Dzik [2] 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 . A partial solution is provided.
Keywords axiom of choice  consequence operation  deductive theory  Dzik system  equilateral consequence operation  extremal consequence operation  Lindenbaum  punctiliousness  saturated theory
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