Arithmetic Foundations are Tautological and Circular

[Eodnhoj7]

As to my point about how the formal proof of one system tends to assume another logic on the 'first order' level, this tends to be 'circular' because one system requires some prior metalogic to prove the logic you are 'testing' and thus cannot necessarily be closed with respect to their dependency upon a collection of systems.

For instance, to prove the Propositional Calculus, you usually need some prior assumption of the validity of a set of boolean truth-functioning tables. Formal Zermelo-Frankel Set theory relies on assuming Predicate logic which is itself dependent upon Propositional logic to prove 'closed' (i.e. 'complete'). Boolean algrebra requires trusting Set theory. Boolean systems include 'multivariables' but requires the first stage is to assume a binary 'set' of values, for instance, and a 'set' of operations declared to be used.

So my question is which metalogic system is most basic an doesn't require another system of logic ASSUMED in order to prove it complete?

There isn't one, all contexts are assumed and context is inescapable, this thread addresses this:

Highest Abstraction for Logic and Math?
viewtopic.php?f=26&t=28138

I said I do not follow your own language or thought process so will not participate further than my comment about metalogic 'circularity'. Thank you.

Then just draw it out by hand, the proof is in the counting of the phenomenon.