Self-verifying theories are consistent first-order systems of arithmetic much weaker than Peano arithmetic that are capable of proving their own consistency. Dan Willard was the first to investigate their properties, and he has described a family of such systems. According to Gödel's incompleteness theorem, these systems cannot contain the theory of Peano arithmetic, and in fact, not even the weak fragment of Robinson arithmetic; nonetheless, they can contain strong theorems. sentence expressing totality of multiplication: where is the three-place predicate which stands for We can add any true