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.
- Monadic second order logic of one successor
- Ordered additive group of real numbers
- Ostrowski numeration system
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