# Thread: Model theory - definiable sets 2

1. ## Model theory - definiable sets 2

Here is more difficult exercise (for me of course) from definiable sets:

Let $\displaystyle A$ be the partial ordering (in a signature with $\displaystyle \leq$) whose elements are the positive integers, with $\displaystyle m \leq ^A$ iff $\displaystyle m$ divides $\displaystyle n$.
Show that the set $\displaystyle \{1\}$ and the set of primes are both $\displaystyle \emptyset$-definable in $\displaystyle A$.

I don't have any idea how can I solve it. Could You give me some advices?
Any help will be highly appreciated.

2. Well, 1 is the only positive integer that divides all other positive integers. And a number other than 1 is prime if its divisors are 1 and itself. It's easy to write these properties as formulas.

3. Could You show me how can I do this, for example in second case (set of primes)?
This is probably easy but I'm really newbie in this subject so I'll be glad if You write some more details. Thanks.

4. $\displaystyle \mathrm{prime}(x)$ is $\displaystyle \neg\mathrm{one}(x)\land\forall y.\,y\le x\leftrightarrow(\mathrm{one}(y)\lor y=x)$. Here $\displaystyle \mathrm{one}(y)$ means $\displaystyle y=1$ (the first part of the problem). I also assume that = is in the language and is interpreted as identity.