On two systems for arithmetic
Toshio Nishimura, Hisao Tanaka · Journal of the Mathematical Society of Japan · 1965
7, $\vee,$ $\wedge,$ $\exists,$ $\forall$ ; the equality symbol $=$ ; the symbol $r$ ; and the parentheses $(, )$ .2.2.We shall correlate distinct odd numbers to the primitive symbols, thus: 5 9 11 13 15 17 19 21 23 25 27, respectively.2.3.Terms are defined inductively and correlated the G\"odel numbers (denoted by rtl for a term t) in the following manner:(1) $0$ and $t\}_{j}$ are terms, and $ro\urcorner=3$ and $r_{t)_{j}}\urcorner=7^{j+1}$ .(2) If $s$ and $t$ are terms, then $t^{\prime},$ $t^{\backslash },$ $s+t,$ $s\cdot t$ and $s^{t}$ are terms and $\Gamma t^{\prime} eg=2^{5}\cdot 3^{\lceil t1}$ , $[t^{\backslash } eg=2^{9}\cdot 3^{\lceil t\rceil},$ $\Gamma s+t eg=2^{11}\cdot 3^{\lceil S\rceil}\cdot 5^{\lceil t\rceil}$ , $rs$ .tl $=2^{13}\cdot 3^{\lceil S\rceil}\cdot 5^{\lceil t\rceil}$ and $[s^{t} eg=2^{15}\cdot 3^{\lceil s\rceil}\cdot 5^{\Gamma^{t\rceil}}$ .