**Followup to**: Where Recursive Justification Hits Bottom, Löb’s Theorem

Peano Arithmetic *seems *pretty trustworthy. We’ve never found a case where Peano Arithmetic proves a theorem T, and yet T is false in the natural numbers. That is, we know of no case where []T ("T is provable in PA") and yet ~T ("not T").

We also know of no case where first order logic is invalid: We know of no case where first-order logic produces *false conclusions* from *true premises*. (Whenever first-order statements H are true of a model, and we can syntactically deduce C from H, checking C against the model shows that C is also true.)

Combining these two observations, it seems like we should be able to get away with adding a rule to Peano Arithmetic that says:

All T: ([]T -> T)

But Löb’s Theorem seems to show that as soon as we do that, everything becomes provable. What went wrong? How can we do *worse* by adding a true premise to a trustworthy theory? Is the premise not true – does PA prove some theorems that are false? Is first-order logic not valid – does it sometimes prove false conclusions from true premises?

Continue reading "You Provably Can’t Trust Yourself" »

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