Abstract
In the language lbrace 0, 1, circ , preceq rbrace , 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} and {textsf {D}}, that are mutually interpretable with the theory of arithmetic {textsf {R}} and Robinson arithmetic {textsf {Q}} , respectively. The intended model of textsf {WD} and {textsf {D}} is the free semigroup generated by lbrace {varvec{0}}, {varvec{1}} rbrace under string concatenation extended with the prefix relation. The theories textsf {WD} and {textsf {D}} are purely universally axiomatised, in contrast to {textsf {Q}} which has the varPi _2-axiom forall x ; [ x = 0 vee exists y ; [ x = Sy ] ] .
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