Consider the signature of arithmetic (+, ·, =) and its natural structure on the set N. Prove that there exists a triple relation S(x, a, b) expressible in this signature, which simultaneously has the following properties: ((a) for any a and b, the set Sab = {x | S(x, a, b)} of course; (b) among the sets of Sab, for different pairs a, b, all finite sets occur
In addition to this task, there is a remark that it is necessary to make a bijection between natural numbers and binary sets of sets (I suppose this is just a translation into binary), then build the relation "the word x is a concatenation of the words y and z" and then take as $S_{ab}$"x is shorter than a and axa is a subword of the b word."
I want to express "the word x is a concatenation of the words y and z" as follows: $\exists k \exists m \exists t (x =y+kz\land (k=2^{m})\land (k = x +t))$ , but here there is a problem that I am not be able to express $2^{m}$, including this predicate is also required to express further expressions, is it possible to express it and how? Perhaps I have approached this task incorrectly or there is a more rational approach to this task