Key Takeaway: The theory of (R,, +, Z, Za) is decidable if a is quadratic and defines multiplication by a if a is the golden ratio.

Abstract

Abstract The theory of (ℝ, <, +, ℤ, ℤa) is decidable if a is quadratic. If a is the golden ratio, (ℝ, <, +, ℤ, ℤa) defines multiplication by a. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of a to define the above structures in monadic second order logic of one successor. The converse that (ℝ, <, +, ℤ, ℤa) defines monadic second order logic of one successor, will also be established.