Prove or disprove if it is enumerable I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah and I have to prove or disprove this: If $C$ is an inductive structure with a finite number of blocks and a finite number of operations, then the number of elements in C is enumerable. What happens if the number of blocks is enumerable? Thanks for your help.

This is not a solution but hints that should help you finding the solution by yourself. If it does not please ask.

First case: finite number of block + finite number of operation: you are right $C$ is enumerable. Hints to prove it:

What can you say about the number of element obtained with less than $n$ operations?

Can you use this to enumerate all the element of $C$?

> hint: if $e\in C$ then there exists $n$ such that $e$ is obtained by applying less than $n$ operations

Second case: enumerable set of blocks + finite rules: $C$ is still enumerable. Hints to prove it:

Let the blocks be $\\{B_1,B_2,\dots\\}$.

What can you say about the number of elements obtain with less than $n$ operation **and** only with blocks form $\\{B_1,\dots,B_n\\}$?

Can you use this to enumerate all the element of $C$?

> Again if $E\in C$ then there exist $n$ such that $e$ is obtained with less than $n$ operation **and** only with blocks form $\\{B_1,\dots,B_n\\}$