Whats wrong with this model theoretic proof of the twin primes conjecture?

Question

I have a proof of the twin primes conjecture using the compactness theorem. It cannot be correct, because it is too simple. Please help find the flaw.

Proof by contradiction, Assumption: there are only $n$ twin primes.

Let $L$ be the language $\{ +, \cdot, 0, S \}$ and $T = Th(\mathbb{N},L)$ be the first order theory of $\mathbb{N}$, with the standard interpretation of the symbols in $L$, with $S$ being the successor function. In the following, numbers like $1$, $2$, $3$ etc. in formulas represent the constant terms $S0$, $SS0$, $SSS0$ etc.

Now let $L^*$ be the language $L$, augmented by $n+1$ constants:

$L^* = L \cup \{ c_{i}\mid i \in N\}$, with $N = \{ 1, ..., n+1\}$

Consider the following sets of formulas:

$\Phi_1 = \{ \neg \mathrm{divides}(p,c_i) \land \neg \mathrm{divides}(p,c_i+2) | i \in N, p \in Primes \}$

$\Phi_2 = \{ c_i \not= c_j \mid i,j\in N, i \not= j \}$

Here $Primes$ denotes the set of constant terms that represent primes (i.e. $P=\{2,3,5,... \}$), and $\mathrm{divides}(p,c)$ is the formula $ \exists k(k \not= 1 \land k \not=c \land k \cdot p = c)$, asserting that $p$ is a true divisor of $c$.

Then $\Psi = T \cup \Phi_1 \cup \Phi_2$ is inconsistent. By compactness, there must be an inconsistent finite subset $\Psi_0 \subset \Psi$. In particular, $\Psi_0$ can only contain finitely many formulas of $\Phi_1$, referencing only finitely many primes. Let $M \subset Primes$ be the set of primes being referenced, and $\Pi M$ the product of all primes in $M$. Then for any $i\ge 1$, the numbers $i\cdot \Pi M-1$ and $i\cdot \Pi M+1$ are coprime to M. Thus interpreting every $c_i$ with $i\cdot \Pi M-1$, we have an interpretation that satisfies all formulas in $\Psi_0$, contradiction.

Answer 1

It is not correct to conclude that $\Psi$ is inconsistent. It is true that there is no assignment of values to the $c_i$ in the standard model $\mathbb{N}$ which makes it a model of $\Psi$. However, that doesn't mean that $\Psi$ is inconsistent, because there could be some different model of it (with a nonstandard underlying model of natural numbers). Indeed, your argument proves $\Psi$ is finitely satisfiable, so by compactness such a model exists.

Here is a simpler example of the same phenomenon. Add just a single constant $c$ to the language of arithmetic and consider sentences $\varphi_n$ saying that $c>n$ for each $n\in\mathbb{N}$. Then $T\cup\{\varphi_n:n\in\mathbb{N}\}$ is finitely satisfiable, but again there is no way to give $c$ a value in $\mathbb{N}$ to make it a model. Instead, to get a model you need to take a nonstandard model of $T$ which has elements which are greater than every standard natural number.

Answer 2

In fact, not only is your theory consistent, but (regardless of the status of the twin primes conjecture) in any nonstandard model of arithmetic we can find interpretations of the $c_i$ (we can even do that with countably many $c_i$, not just $n+1$).

The point is that the factorial function makes sense in any nonstandard model of arithmetic, using Gödel's argument to code recursive functions within the language of arithmetic, see here. Let $M$ be a nonstandard model of arithmetic and let $I$ be an infinite (i.e., nonstandard) number in $M$. For $i\in N$, let $c_i=(I+i)!-1$. The $c_i$ so defined are distinct, and the usual (Euclid's) argument for the infinitude of primes shows that no standard prime divides any of the $c_i$ or of the $c_i+2$. Note that absolutely nothing changes if we have countably many $c_i$ rather than only $n+1$.

(Now, if the twin primes conjecture is false, and $N$ as in your post, then in no model of your theory is any of the $c_i$ a prime number. But this is not an issue.)