> NA (The Nathan's Axiom): All sets are finitely constructible. > > Theorems of ZF + NA are not the same ones from ZF alone. From ZF + NA, > you can deduce that there are only countably many reals (since most > transcendental reals will not exist in ZF + NA). However, this > statement is false in ZFC.
I think that ZF + NA is inconsistent, since NA contradicts the Axiom of Replacement. All those sets he created S_i for finite integer i should be replaceable with any ordinal i, creating sets of infinite constructibility.
Even in Nathan's own unstated axioms, his world is inconsistent with ZF.
Note that this isn't a problem in the Constructible universe of V=L, since Nathan's axiom does not hold, and there are still uncountably many reals.