Original version
Archive for Mathematical Logic. 2022, DOI: https://doi.org/10.1007/s00153-022-00820-y
Abstract
In the language {0,1,∘,⪯}, where 0 and 1 are constant symbols, ∘ is a binary function symbol and ⪯ is a binary relation symbol, we formulate two theories, WD and D, that are mutually interpretable with the theory of arithmetic R and Robinson arithmetic Q, respectively. The intended model of WD and D is the free semigroup generated by {00,11} under string concatenation extended with the prefix relation. The theories WD and D are purely universally axiomatised, in contrast to Q which has the Π2-axiom ∀x[ x=0∨∃y[ x=Sy ] ].