A Note on an Arithmetization of a Word System in a Denumerable Alphabet

Abstract

This note presents a new arithmetization of a word system in a denumerable alphabet, based on a certain recursively definable class of natural numbers, introduced by JAN MYCIELSKI [l], which is an extension of the class of consecutive prime numbers.Throughout this note we employ the following notation: m n (a modified BOUR-BAKI notation for negation) ; A (conjunction) ; v (disjunction) ; *+ (equivalence);A (universal quantifier) ; V (existential quantifier) ; p (operation of minimalization). We shall also have occasion to use the following equality relation for sequences of natural numbers :E ( x , y; 1,1).+*x, = y1, E ( x , y; 1 , n) false for n > 1 , E ( x , y;m, 1 ) false for m > 1, E(x, y; m + 1 , n + 1) +=+ ( z m + i = Y~+ I ) A E ( x , y; m , n)

Commutative recursive word arithmetic in the alphabet of prime numbers.

“…Following the author's earlier paper, 3 we now construct an arithmetical interpretation of the word system V(A), which is exactly the class of all possible Godel numbers of V (A ).…”

Primitive Words in an Infinite Abstract Alphabet

“…(2.11) (2.12) P of words as in [2], we state our k-primitive-word unique resolution theorems: P n q p n$, P, v 1(P) $. 1(P,) v n""pw'k'(P) v P lI(k) P,.…”