(pp. 157-161)

Now we can begin to prove results about predicative arithmetic within predicative arithmetic. In this chapter we will arithmetize Robinson’s theory and show it to be tautologically consistent (and we do this within a theory that is interpretable in Robinson’s theory).

29.1$Def.\tilde 0 = {F_{0,0}}$.

29.2$Def.\bar S = {F_{1,0}}$

29.3$Def.\bar P = {F_{1,1}}$

29.4$Def. \mp = {F_{2,0}}$

29.5$Def.\bar \cdot = {F_{2,1}}$

29.6$Def.\tilde Sa = \bar S * a$

29.7$Def.\tilde Pa = \bar P * a$

29.8$Def.a\tilde + b = \mp * a * b$

29.9$Def.a\tilde \cdot b = \bar \cdot * a * b$

If x is the ν^{th}variable in alphabetical order, we use${\tilde x}$as an abbreviation for${X_{\bar \nu }}$. (Recall that${\bar \nu }$is$S \cdots {S_0}$with ν occurences of S.) Also, we use$\{ {a_1}, \ldots {a_\nu }\} $as an abbreviaion of$\{ {a_1}\} U \cdots U\{ {a_\nu }\} $.

29.10$Def.{{\bar Q}_0} = (\{ \tilde 0,\bar S,\bar P, \mp ,\bar \cdot \} ,\{ \tilde S\tilde x\tilde \ne \tilde 0,\tilde S\tilde x\tilde = \tilde S\tilde y\tilde \to \tilde x\tilde = \tilde y,\tilde x\tilde + \tilde 0\tilde = \tilde x,\tilde x\tilde + \tilde S\tilde y\tilde = \tilde S(\tilde x\tilde + \tilde y),\tilde x\tilde \cdot \tilde 0\tilde = \tilde 0,\tilde x\tilde \cdot \tilde S\tilde y\tilde = \tilde x\tilde \cdot \tilde y\tilde + \tilde x,\tilde P\tilde x\tilde = \tilde y\tilde \leftrightarrow \tilde S\tilde y\tilde = \tilde x\tilde V(\tilde x\tilde = \tilde 0\tilde \& \tilde y\tilde = \tilde 0)\} )$

29.11*Def. a*...