Reading Hodges, I encountered two definitions of types of a theory. I think I really grasped the concept, so that either definition he uses I can uderstand proofs. But nonetheless I cannot fully see the equivalence between these two definitions and that is disturbing to me.

In section 2.3 we have the following:
Definition 1 A n-type of a theory $T$ is a collection of formulas $\Phi(\bar{x})$ such that there is a model $A$ of $T$ and a n-tuple $\bar{a}$ from $A$ such that $A\models\phi(\bar{a})\quad\forall\phi\in\Phi$. We then say that another model $B$ of the same theory $T$ realizes the type if the same olds in $B$, otherwise we say it omits the type.

In section 5.2 things are handled differently, starting from structures instaed of theories:
Let $A$ be a structure and $\bar{b}$ an n-tuple of elements from $A$. The complete n-type of $\bar{b}$ over the set of parameters $X\subset A$ with respect to $A$ is collection $\Phi(\bar{x},\bar{y})$ of formulas such that $A\models\phi(\bar{b},\bar{a})\quad\forall\phi\in\Phi$ for some $\bar{a}\subset X$. Then we say that a complete n-type is such a collection of formulas but allowing for some $\bar{b}$ which lies in some elementary extension $B$ of $A$. In the parricular case where $\bar{b}$ is actually in $A$ we say $A$ realizes the type otherwise we say it omits the type. A type of $A$ is simply a subset of some complete type of $A$.

Thanks to the compactness theorem, we can show that a collection of formulas $\Phi(\bar{x})$ is a type of $A$ if and only if $A\models\exists\bar{x}\bigwedge\Psi(\bar{x})$ where $\Psi(\bar{x})$ varies in finite subsets of $\Phi(\bar{x})$. Similarly a maximal collection of formulas with this property is the same thing as a complete type of $A$.

With this background we say that
Definition 2 A type of a theory T is a collection $\Phi(\bar{x})$ of formulas such that $T'=T\cup\{\exists\bar{x}\bigwedge\Psi(\bar{x})\}$ is consistent for any finite subset $\Psi(\bar{x})$ of $\Phi(\bar{x})$

I cannot see why these two are equivalent. In parricular 1 seems stronger to me. I would try to show that 2 implies 1 by the compactness theorem and the equivaent definition of type in a structure given above. But no one assures me that the different structures witnessing the consistency of $T'$ can be put together in a single one.

Thanks in advance for help

P.s. Notice that Equivalent Definitions of Types is a different question.


1 Answer 1


No one assures me that the different structures witnessing the consistency of $T'$ can be put together in a single one.

That's exactly what the compactness theorem allows you to do.$^*$ Perhaps the following argument will be more transparent:

Extend the language with constant symbols $\bar c$. Let $T''$ be $T$ plus $\phi(\bar c)$ for all $\phi\in \Phi.$ Then the condition implies $T''$ is finitely satisfiable: if $\Psi$ is the finite set of axioms used, take a model of $T\cup \exists \bar x\bigwedge \Psi(\bar x)$, and then assign $\bar c$ to the witness to $\exists \bar x\bigwedge \Psi(\bar x).$ Thus by compactness, $T''$ has a model. In this model, whatever $\bar c$ is interpreted as will realize the type $\Phi.$

$^*$This is a somewhat misleading way of putting it. You aren't actually putting the structures together. You are, through the magic of the compactness theorem, coming up with an entirely different structure from scratch.

For instance, oftentimes, each of the finite subtheories is satisfied by the same base structure (with different relevant symbols and assignments for them), but then the model that exists by compactness is something novel.

As an example, recall the compactness argument that any satisfiable theory with infinite models has models of arbitrarily large cardinality. We don't amalgamate the models we find for the finite subtheories (which are all the same size if we're doing things in the most straightforward way) to get the large model... it's just not how the argument works.

  • $\begingroup$ Of course, you're argument is very clear and a quite standard application of the compactness theorem, somehow I was missing the fact that $\Psi$ varies among all finite subsets of the type... In my mind I was thinking about a partition. Thanks a lot, sometimes one just shuts down his own brain $\endgroup$ May 24, 2020 at 10:13
  • $\begingroup$ And that was the reason I was thinking about a kind of amalgam $\endgroup$ May 24, 2020 at 10:20

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