Nonconcatenative Abstract Skolem Arithmetics I

✍ Scribed by H. A. Pogorzelski

• Book ID: 102487251
• Publisher: John Wiley and Sons
• Year: 1965
• Tongue: English
• Weight: 246 KB
• Volume: 11
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19650110203

✦ Synopsis

Skolem arithmetic is an arithmetic constructible on an abstract word system in a finite or denumerably infinite abstract alphabet by means of propositional calculus, definition by composition and recursion in the word system, and proof by induction in the word system, without the use of unbounded quantifiers.

Using versions of RABIN'S [2] nonconcatenative operations, a,, @ a, = a,, + , and a o a , = a,'., (a,,, a,€ A ; p , v E N), where A = {a,, a,, a 3 , . . .}, N = (1, 2 , 3 , . . .} and p + v and pa v are ordinary operations of addition and multiplication, we first construct an abstract SKOLEM arithmetic on the set A itself, and in turn we extend this arithmetic to an abstract SKOLEM arithmetic on the noncommutative word system Q(A) in the alphabet A [4], constructing i t up through its multiplicative unique word resolution theorem.