Within your logic paradigm if an axiom is removed a sub-set
of axioms ensues. Using this, sub-sets of logical constructs can be generated.
Anyone using this smaller domain will be incapable of comprehending, intrinsically, higher contsructs.
Your logical argument suffers from this ailment.
I am an pantheist/agnostic/atheist depending on my mood.
But surely a mathematical proof alone can resolve this.
Words clutter the thought processes.
nk