CRGreathouse wrote:So if the regularity axiom isn't a part of your logic you can't assume it
The proof of consistency is external to the system instance, this holds independently from regularity, and consistency of any instance is shown independently from regularity. Regularity was just used to further document the fact that the same proof cannot become an internal proof, but this also holds by using goedel second incompleteness theorem by contraposition.
There are models (i.e. instances) of my system that don't use regularity, and they are consistent because all models are consistent.
I consider the topic about consistency is clarified. I'll probably not repost about this, since I don't see any consistency issue about my system.
As a further reference on this topic i can cite this part of wikipedia's article about goedel's incompleteness theorems.
The second incompleteness theorem only shows that the consistency of certain theories cannot be proved from the axioms of those theories themselves. It does not show that the consistency cannot be proved from other (consistent) axioms. For example, the consistency of the Peano arithmetic can be proved in Zermelo–Fraenkel set theory (ZFC), or in theories of arithmetic augmented with transfinite induction, as in Gentzen's consistency proof.