Provider: JSTOR http://www.jstor.org Database: JSTOR Content: text/plain; charset="us-ascii" TY - CHAP TI - The arithmetization of arithmetic A2 - Nelson, Edward AB -

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

29.11Def. a

EP - 161 PB - Princeton University Press PY - 1986 SN - null SP - 157 T2 - Predicative Arithmetic. (MN-32): UR - http://www.jstor.org/stable/j.ctt7ztgtr.31 Y2 - 2020/12/02/ ER -