User blog comment:MilkyWay90/Math number/@comment-35392788-20180902214346/@comment-35470197-20180902230351

Since it partially defines syntax, it is not so difficult to avoid the paradox, as long as the OP defines what "recursion" precisely means here. One of the most problematic fact is that no axioms are declared, and hence the "well-definedness" of an expression N in the definition of Math(x) is not well-defined. (The well-definedness heavily depends on axioms, especially on the induction schema.)

If the OP considers Peano arithmetic or some othe rarithmetic, then the resulting function Math(x) is uncomputable, because there is no recursive way to determine whether a given expression N is well-defined under the arithmetic or not. So this is kind of a weakened Busy Beaver function.