I don't know what you mean by "minimal inconsistency guard". — TonesInDeepFreeze
specialized sense of consistency ― that P and ~P are inconsistent, for any P — Srap Tasmaner
you can say A and B are inconsistent if A → C and B → ~C. — Srap Tasmaner
unless you want to start with the Sheffer stroke — Srap Tasmaner
we need both consistency and consequence as core ideas — Srap Tasmaner
any given idea (claim, thought, etc.) has a twin that is the one thing guaranteed under no circumstances to follow from it — Srap Tasmaner
We have that. — TonesInDeepFreeze
We already have: — TonesInDeepFreeze
We define consistency from provability. — TonesInDeepFreeze
Why is that lacking? — TonesInDeepFreeze
How do you know there is only one thing? — TonesInDeepFreeze
Set theory is needed for the rest of math and so is logic — Srap Tasmaner
let the negation of C(P) be N(P) — TonesInDeepFreeze
Much of classical math existed before the introduction of set theory. — jgill
Might be interesting to adduce a formal sentence and demonstrate somehow that it can't be said in English alone (not just that all known attempts failed). — TonesInDeepFreeze
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