0
$\begingroup$

I'm working my way through Enderton and am stuck on the following exercise.

Take a language containing $\forall$, $=$, and the binary predicate $P$. Define the model $\frak{M}$ so that $\frak{M}= \mathbb{Z}$ and $(x,y)\in P^{\frak{M}}$ iff $|x-y|=1$.

I want to show that there exists some model $\frak{N}$ that is elementarily equivalent to $\frak{M}$ and that is not connected. Enderton defines connected as follows. For any two $a,b \in \frak{M}$, there is a path between them, i.e., a sequence $(p_0, p_1, \ldots, p_m)$ so that $p_0 =a$ and $p_m =b$ and $(p_j, p_{j+1}) \in P^{\frak{M}}$ for each $j$. Also, as usual, elementary equivalence means $\models_{\frak{N}} s$ iff $\models_{\frak{M}} s$ for any sentence $s$ in the language.

I'm given a suggestion to add two constant symbols to the language and write a sentence saying that they are "far apart". Then I apply compactness.

I honestly have no idea why this suggestion will work. Ultimately, I'm curious to see how this exercise is proved exactly. Also, I don't know any technical graph theory, so I'd appreciate answers that omit sophisticated terminology of that kind.

1 Answers 1

2

Remember that compactness says:

If $\Gamma$ is a set of sentences, and every finite subset of $\Gamma$ has a model, then $\Gamma$ has a model.

Our goal in an exercise like this is to do the following: find a set of sentences (in a larger language) which contain the theory of $\mathcal{M}$ (so any model's reduct to the original language will be $\equiv\mathcal{M}$), but which also include sentences describing a phenomenon which does not happen in $\mathcal{M}$.

For example, if $\mathcal{M}$ is $(\mathbb{N}; 1, +)$, a natural way to do this is to expand our language by a new constant $c$, and consider the set of sentences $\Gamma$ consisting of

  • All sentences in the original language true of $\mathcal{M}$, and

  • The sentence "$\neg(1+1+...+1=a)$" ($n$ $1$s) for every $n\in\mathbb{N}$.

Then:

  • Any finite subset $\Delta$ of $\Gamma$ has a model, in particular some expansion of $\mathcal{M}$; just make $a$ denote an element larger than any of the (finitely many!) $n$s such that "$\neg(1+1+...+1=a)$" ($n$ $1$s) is in $\Delta$.

  • But $\mathcal{M}$ cannot be expanded to a model of all of $\Gamma$ - where could $a$ go?

So while $\Gamma$ has a model by Compactness, the reduct to the original language of any such model will be a structure elementarily equivalent to, but not isomorphic to, $\mathcal{M}$.

In the example you give, the phenomenon we want to add is two points that are infinitely far apart. To do this, we'll expand our language by two new constant symbols, $a$ and $b$, and add to the theory of $\mathcal{M}$ the sentences

  • $\neg\exists x_1, . . ., x_n(P(a, x_1)\wedge P(x_1, x_2)\wedge . . . \wedge P(x_n,b))$ (that is, $a$ and $b$ are at least $n$ edges apart) for each $n\in\mathbb{N}$.

Do you see why this collection of sentences is finitely consistent? And do you see why any model of this collection of sentences must have an underlying graph which is not isomorphic to $\mathcal{M}$?

  • 0
    This is great!. Just one clarification: doesn't the sentence $\neg\exists x_1, \ldots, x_n(P(a, x_2) \land P(x_1, x_2)\land \ldots \land P(x_n, b))$ mean here "there is no path of length $n$ between $a$ and $b$"? (Which cannot be true in a model in the original language for all of $\Gamma$). – 2017-02-23
  • 1
    @CuriousKid7 That's what I meant when I said $a$ and $b$ are at least nn edges apart. And it's not true that this can't be true in a(n expansion of a) model in the original language for all of $\Gamma$ - it just can't be true for any expansion of *the specific model $\mathcal{M}$.* The whole point of Compactness is that it *can* be true in some model. – 2017-02-23
  • 0
    Right that all makes sense. Thanks again! – 2017-02-23
  • 0
    One more question: Do we need to add the sentence for each natural $n$, or would it suffice to just add one sentence saying that $a,b$ are at least, say, $5$ edges apart? wouldn't the collection still be finitely consistent? – 2017-02-24
  • 0
    @CuriousKid7 As long as you have enough sentences - you need to say that they're at least $n$ apart *for arbitrarily large $n$* (or else you don't necessarily get something different from your starting structure). – 2017-02-24