Let $A\subset\mathbb{N}$ "finally cyclic" (I give the definition below). Show, that $A$ is definable over $(\mathbb{N},+)$
Hello,
I have a question to this task. I have to show, that the set $A$ is definable over $(\mathbb{N},+)$. So I have to give a formula $\varphi$ which describes the set. Or more specific, which describes every element of this set.
Definition:
$A\subset\mathbb{N}$ is "finally cyclic" iff $\exists n_0\exists m_0\forall n\geq n_0(n\in A\Leftrightarrow n+m_0\in A)$.
The problem of this task is, that I have to describe every element of this set. For example:
$A=\{1,7,29,30,33,35,100,101,102,103,104, ...\}$ is finally cyclic, because for $n_0=100$ exists $m_0=1$ and every number which is greater than 100 is an element of $A$.
To give the part of the formula, which describes the numbers greater than 100 is no problem. But what is with the numbers 1, 7, 29, 30, 33, 35?
Well, it is clear, that the amount of "non-cyclic-numbers" is finite. So you can just write them down one by one, with variables. Like $v_1, v_2, v_3, v_4, v_5$ and $v_6$. Or $v_1, ..., v_i$ in general. With $v_1=1, ..., v_6=35$
Another problem is to show, that the formula really describes a set. That every element is just once in the set.
Also I am not sure, how a formula can describe a set in generall. When I want to define the random set $B=\{0,1\}$ over $(\mathbb{N},+,\cdot)$.
Is it enough to describe those two numbers, and does this make the formula already a "set", when it describes a list of numbers (or what so ever)?
Can someone help me with my questions? Thanks in advance.
I sum up partial answers in the comments to a full answer.
$x\le y$ is definable by $\exists z\ x+z=y$
$x=0$ is definable by $\forall y\ x\le y$
$x=1$ is definable by $0<x\wedge\neg\exists y\ 0<y<x$
$x=m$ for any natural number $m$ is definable.
Therefore, every finite set is definable.
The term $nx$ is definable by $x+\dots(n$ times$)\dots+x$
The range of an arithmetic progression is a definable set: $\exists y\ \ m+ny=x$
Finally, note that definable sets are closed under boolean operations and obtain that any finally cyclic sets is definable.