Models in arithmetic

Question

I have to prove that there exists model of complete arithmetic $\bf A$ such that it contains $a\in A$ and $a$ is divisible by all prime numbers. I don't know where to start.

Answer 1

As developed in the comments, the solution is to first introduce one or more constant symbols standing for the "special" elements we care about, and then break our "big" requirement into many "small" requirements.

In this case, we want to introduce a single constant symbol $c$ to be our thing which is divisible by every prime. Our "big" requirement is "$c$ is divisible by every prime," so our "small" requirements will be each of the sentences "$p$ divides $c$" for $p$ a prime. A bit more precisely, we're looking at the set of sentences $$X=\{\exists x[x\cdot (1+1+...+1\mbox{ ($p$ times}))=c]: p\mbox{ prime}\}.$$ For example, $X$ contains the sentences "$\exists x[x\cdot (1+1)=c]$," "$\exists x[x\cdot (1+1+1+1+1)=c]$," and so forth.

Now consider $TA\cup X$ (where $TA$ is the true theory of arithmetic). It's not hard to show (exercise) that $TA\cup X$ is finitely satisfiable; by compactness, that means it has a model. And any $M\models TA\cup X$ - or rather, the reduct of any such $M$ to the language of arithmetic (just forget the new constant symbol) - is an example of what we want.