Weak essentially undecidable theories of concatenation

Abstract

AbstractIn the language $$\lbrace 0, 1, \circ , \preceq \rbrace $$ { 0 , 1 , ∘ , ⪯ } , where 0 and 1 are constant symbols, $$\circ $$ ∘ is a binary function symbol and $$\preceq $$ ⪯ is a binary relation symbol, we formulate two theories, $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D , that are mutually interpretable with the theory of arithmetic $$ {\textsf {R}} $$ R and Robinson arithmetic $${\textsf {Q}} $$ Q , respectively. The intended model of $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D is the free semigroup generated by $$\lbrace {\varvec{0}}, {\varvec{1}} \rbrace $$ { 0 , 1 } under string concatenation extended with the prefix relation. The theories $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D are purely universally axiomatised, in contrast to $$ {\textsf {Q}} $$ Q which has the $$\varPi _2$$ Π 2 -axiom $$\forall x \; [ \ x = 0 \vee \exists y \; [ \ x = Sy \ ] \ ] $$ ∀ x [ x = 0 ∨ ∃ y [ x = S y ] ] .