We know that there are provability predicates, that is, predicates derived from the recursive relation "x is a demonstration of y", with which Godel's second incompleteness theorem would not be satisfied.
It seems that to prove Godel's second incompleteness theorem we need the provability predicate to satisfy the Hilbert–Bernays provability conditions. But how do we know that the expression of consistency in the system itself is done through a probability predicate that meets those conditions?.
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