Unary-arity obstruction for matching μ-logic

Determine whether the non-effective-completeness result for definedness-free matching μ-logic can be obtained using only unary symbols.

Background

The non-recursive-enumerability theorem uses one unary symbol and two binary symbols. The binary symbols encode addition and multiplication as ternary relations represented at a point, so the proof does not minimize the arity of the signature.

The paper notes that a relativized universal-quantifier mechanism might constrain an encoded triple structure even with unary symbols, but no such construction is provided.