Direct Proofs of Lindenbaum Conditionals

We discuss the problem raised by Miller (Log Univers 1:183–199, 2007 ) to re-prove the well-known equivalences of some Lindenbaum theorems for deductive systems (each equivalent to the Axiom of Choice) without an application of the Axiom of Choice. We present five special constructions of deductive systems, each of them providing some partial solutions to the mathematical problem. We conclude with a short discussion of the underlying philosophical problem of deciding, whether a given proof satisfies our demand that the Axiom of Choice is not applied.