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It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exist, as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I mean is that almost all natural example of theories will be classified with a topos having enough point.

There are two fairly similar theorem that guarantee this:

1. Deligne completness theorem assert that "every Coherent topos has enough points". It implies that any "coherent theory" (that is a geometric theory with not infinite join) is classified by a topos with enough points.

2. Makkai-Reyes completness theorem assert that a "separable topos", i.e. a topos that admits a countable site with a countable set of basic covering famillies, has enough point. (see theorem 6.2.4 in Makkai and Reyes "First Order categorical logic"). It implies that the classyfing topos of a "countable" geometric theory has enough points. By countable I mean that the theory has a countable signature and a countable set of axioms that uses at most countable joins.

It seems to me that any of these two results already cover all the example mentioned in Beke's paper.

To come back to my initial point, to construct a Geometric theory that do not have enough model, you need to start with some uncountable set of data, for example start from an uncountable set $I$ and look at the theory of surjection $\mathbb{N} \to I$. But pretty much all the theories we want to look at in every day mathematics are countable.

Note: Regarding your comment on example 3.3, the point is that he only considers propositional theories whose axiom are of the form $\forall x, \phi(x)$ where $\phi$ has no quantifier. This is very far from being all propositional theories.

It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exist, as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I mean is that almost all natural example of theories will be classified with a topos having enough point.

There are two fairly similar theorem that guarantee this:

1. Deligne completness theorem assert that "every Coherent topos has enough points". It implies that any "coherent theory" (that is a geometric theory with not infinite join) is classified by a topos with enough points.

2. Makkai-Reyes completness theorem assert that a "separable topos", i.e. a topos that admits a countable site with a countable set of basic covering famillies, has enough point. (see theorem 6.2.4 in Makkai and Reyes "First Order categorical logic"). It implies that the classyfing topos of a "countable" geometric theory has enough points. By countable I mean that the theory has a countable signature and a countable set of axioms that uses at most countable joins.

It seems to me that any of these two results already cover all the example mentioned in Beke's paper.

To come back to my initial point, to construct a Geometric theory that do not have enough model, you need to start with some uncountable set of data, for example start from an uncountable set $I$ and look at the theory of surjection $\mathbb{N} \to I$. But pretty much all the theories we want to look at in every day mathematics are countable.

Note: Regarding your comment on example 3.3, the point is that he only considers theories whose axiom are of the form $\forall x, \phi(x)$ where $\phi$ has no quantifier. In the case of propositional theories, it means axiom of the form $\phi$ (with $\phi$ a geometric proposition) instead of the more general geometric sequent $\phi \Rightarrow \psi$ that you can use in a propositional theory.

It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exists as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I means is that almost all natural example of theories will be classified with a topos having enough point. There are two fairly similar theorem that guarantee this:

1. Deligne completness theorem assert that "every Coherent topos has enough points". It implies that any "coherent theory" (that is a geometric theory with not infinite join) is classified by a topos with enough points.

2. Makkai-Reyes completness theorem assert that a "separable topos", i.e. a topos that admits a countable site with a countable set of basic covering famillies, has enough point. (see theorem 6.2.4 in Makkai and Reyes "First Order categorical logic"). It implies that the classyfing topos of a "countable" geometric theory has enough points. By countable I mean that the theory has a countable signature and a countable set of axioms that uses at most countable joins.

It seems to me that any of these two results cover all the example in Beke's paper ?

To come back to my initial point, to construct a Geometric theory that do not have enough model, you need to start with some uncountable set of data, for example start from an uncountable set $I$ and look at the theory of surjection $\mathbb{N} \to I$. But pretty much all the theories we want to look at in every day mathematics are countable.

Note: Regarding your comment on example 3.3, the point is that he only considers propositional theories whose axiom are of the form $\forall x, \phi(x)$ where $\phi$ has no quantifier. This is very far from being all propositional theories.

It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exist, as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I mean is that almost all natural example of theories will be classified with a topos having enough point. 

There are two fairly similar theorem that guarantee this:

1. Deligne completness theorem assert that "every Coherent topos has enough points". It implies that any "coherent theory" (that is a geometric theory with not infinite join) is classified by a topos with enough points.

2. Makkai-Reyes completness theorem assert that a "separable topos", i.e. a topos that admits a countable site with a countable set of basic covering famillies, has enough point. (see theorem 6.2.4 in Makkai and Reyes "First Order categorical logic"). It implies that the classyfing topos of a "countable" geometric theory has enough points. By countable I mean that the theory has a countable signature and a countable set of axioms that uses at most countable joins.

It seems to me that any of these two results already cover all the example mentioned in Beke's paper.

To come back to my initial point, to construct a Geometric theory that do not have enough model, you need to start with some uncountable set of data, for example start from an uncountable set $I$ and look at the theory of surjection $\mathbb{N} \to I$. But pretty much all the theories we want to look at in every day mathematics are countable.

Note: Regarding your comment on example 3.3, the point is that he only considers propositional theories whose axiom are of the form $\forall x, \phi(x)$ where $\phi$ has no quantifier. This is very far from being all propositional theories.

It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exists as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I means is that almost all natural example of theories will be classified with a topos having enough point. There are two fairly similar theorem that guarantee this:

1. Deligne completness theorem assert that "every Coherent topos has enough points". It implies that any "coherent theory" (that is a geometric theory with not infinite join) is classified by a topos with enough points.

2. Makkai-Reyes completness theorem assert that a "separable topos", i.e. a topos that admits a countable site with a countable set of basic covering famillies, has enough point. (see theorem 6.2.4 in Makkai and Reyes "First Order categorical logic"). It implies that the classyfing topos of a "countable" geometric theory has enough points. By countable I mean that the theory has a countable signature and a countable set of axioms that uses at most countable joins.

It seems to me that any of these two results cover all the example in Beke's paper ?

Note: Regarding your comment on example 3.3, the point is that he only considers propositional theories whose axiom are of the form $\forall x, \phi(x)$ where $\phi$ has no quantifier. This is very far from being all propositional theories.

It is actually not that simple to contruct geometric theories whose classifying toposes do not have enough points. Of course they exists as any Grothendieck topos is the classifying topos of something and there are plenty of Grothendieck topos that do not have enough points. But what I means is that almost all natural example of theories will be classified with a topos having enough point. There are two fairly similar theorem that guarantee this:

1. Deligne completness theorem assert that "every Coherent topos has enough points". It implies that any "coherent theory" (that is a geometric theory with not infinite join) is classified by a topos with enough points.

2. Makkai-Reyes completness theorem assert that a "separable topos", i.e. a topos that admits a countable site with a countable set of basic covering famillies, has enough point. (see theorem 6.2.4 in Makkai and Reyes "First Order categorical logic"). It implies that the classyfing topos of a "countable" geometric theory has enough points. By countable I mean that the theory has a countable signature and a countable set of axioms that uses at most countable joins.

It seems to me that any of these two results cover all the example in Beke's paper ?

To come back to my initial point, to construct a Geometric theory that do not have enough model, you need to start with some uncountable set of data, for example start from an uncountable set $I$ and look at the theory of surjection $\mathbb{N} \to I$. But pretty much all the theories we want to look at in every day mathematics are countable.

Note: Regarding your comment on example 3.3, the point is that he only considers propositional theories whose axiom are of the form $\forall x, \phi(x)$ where $\phi$ has no quantifier. This is very far from being all propositional theories.