Where have we got to in this slow introduction to the incompleteness theorems? Theorem  8 from Chapter 4 tells us that if a consistent, effectively axiomatized, theory is “sufficiently strong”, then it must be negation incomplete. And we announced that even the very weak arithmetic called Robinson Arithmetic is sufficiently strong (in which case, stronger consistent, effectively axiomatized, theories must also meet the condition, and so must be incomplete). But we didn’t say anything at all about what this weak theory of arithmetic looks like! Today’s Chapter 5 explains. So here are revised versions of Chapters 1 to 4 together with the new Chapter 5 on ‘Two weak arithmetics’. (Many thanks to those who have commented on the earlier chapters here or by email: there are quite a few small improvements as a result.) The post Gödel Without Tears, slowly, 5 appeared first on Logic Matters.