Self-Verifying Theories
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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, but they can nonetheless contain strong theorems; for instance there are self-verifying systems capable of proving the consistency of Peano a...
High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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, but they can nonetheless contain strong theorems; for instance there are self-verifying systems capable of proving the consistency of Peano arithmetic. In outline, the key to Willard's construction of his system is to formalise enough of the Gödel machinery to talk about provability internally without being able to formalise diagonalisation. Diagonalisation depends upon being able to prove that multiplication is a total function (and in the earlier versions of the result, addition also).
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