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I have a question about models of a set of sentences $T$, specifically the following:

Let $S=\{R\}$ where $R$ is a unary relation symbol. Let $T$ be the set of sentences that for each $n\geq 1$ contains a sentence saying there are at least $n$ many elements for which $R$ holds and does not hold:

$\exists x_1...x_n(x_0\neq x_1\wedge x_0\neq x_2\wedge...\wedge x_{n-2}\neq x_{n-1})\wedge R(x_1)\wedge R(x_2)\wedge...\wedge R(x_n))$

$\exists x_1...x_n(x_0\neq x_1\wedge x_0\neq x_2\wedge...\wedge x_{n-2}\neq x_{n-1})\wedge \lnot R(x_1)\wedge\lnot R(x_2)\wedge...\wedge\lnot R(x_n))$

Must a substructure of a structure satisfying $T$ also satisfy $T$? Describe all the countable models up to isomorphism and all the models which have size $\aleph_1$ up to isomorphism.

So obviously this is not a question asking for a proof but more an explanation. In this sort of abstract case I can't seem to get a grip on even one model of $T$, any help would be appreciated.

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  • $\begingroup$ @AndréNicolas How are you coming up with those numbers? $\endgroup$ – Jimmy2Goons Apr 7 '16 at 0:12
  • $\begingroup$ I have written an answer with a little more detail. If you have trouble proving one of the assertions, I can supply further help. $\endgroup$ – André Nicolas Apr 7 '16 at 0:21
  • $\begingroup$ @bof Yes sorry typo $\endgroup$ – Jimmy2Goons Apr 7 '16 at 0:44
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For one concrete model of cardinality $\aleph_0$, let the underlying set be the set of natural numbers, and let the interpretation of $R$ hold at $n$ precisely if $n$ is even.

You should be able to show that any countable model is isomorphic to the above model.

There are, up to isomorphism, three models of cardinality $\aleph_1$. They correspond to the following possibilities:

(i) (The interpretation of) $R$ is true at uncountably many places,and false at uncountably many places.

(ii) $R$ is true at uncountably many places, and false at denumerably many places.

(iii) $R$ is true at denumerably many places, and false at uncountably many places.

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  • $\begingroup$ Thanks, this helps a lot. So in regards to $\mathbb{N}$, a substructure like $2\mathbb{N}$ does not satisfy $T$ correct? So it is not true in general that a substructure must satisfy $T$? $\endgroup$ – Jimmy2Goons Apr 7 '16 at 0:23
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    $\begingroup$ Good. A substructure of a model of $T$ need not be a model of $T$. Alternately, we could take the substructure with elements $0,1,2$. $\endgroup$ – André Nicolas Apr 7 '16 at 0:26

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