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The following issue with the use of parameters in model theory has been one of the greatest challenges in my attempt to learn model theory.

Based on what I have seen, I understand that model theorists often do not make explicit the set to which the parameters of formulas and types belong when they know what they are talking about. I have also observed that they do this when they teach model theory.

For instance, consider the following definition of forking the likes of which I have seen in multiple textbooks of model theory:

A formula $\phi(x, a)$ forks over $B$ if $\phi(x, a)$ implies a finite disjunction $\bigvee_i \psi_i(x, b_i)$ of formulas $\psi_i(x, b_i)$ that divide over $B$.

I asked myself, where do the parameters of $\psi_i$ belong? Are they formulas over $B$?

Of course, a model theorist would immediately say that, since no formula over $B$ divides over $B$, the formulas $\psi_i$ are meant to have parameters from elsewhere (anywhere in the monster model?).

I don't know model theory, however, and that is the very reason I read those textbooks.

The question I would like to ask is how I can correctly guess the set from which model theorists take parameters.

Addendum: This is not a question how parameters work in general! This is not a question on definitions of forking either, although I would appreciate comments that clarifies the definition above.

Pteromys
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    Correction: No formula over $B$ divides over $B$. Formulas over $B$ can fork over $B$. – Alex Kruckman Sep 25 '17 at 00:43
  • @AlexKruckman Thank you for pointing this out; that is what I meant to write. – Pteromys Sep 25 '17 at 01:07
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    In the particular case of the definition of forking, notice that the definition of $\psi(x,b)$ divides over $A$ does not require $b$ to be in any particular set, to $b$ can be assumed to be in the monster model. Hence, when it says that $\phi(x,c)$ forks over $A$ if it implies a finite disjunction of formulas $\psi_i(x,b_i)$ that divide over $A$, since there are no restrictions on the tuples of parameters, you might assume they are taken in the monster model too. – Darío G Jan 30 '18 at 07:55
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    More generally, trying to guess where the parameters of formulas are taken from could be difficult. My personal strategy is to assume they are taken from the monster model unless a set of parameters has been mentioned previously and, before continue the reading, check with my toy examples if it makes sense. This can be quite a slow sometimes, but might help you understand better the situation. – Darío G Jan 30 '18 at 08:00
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    When writing, I try my best not to use in the same paragraph a lower-case letter name for parameters that are not in the same upper-case letter set (i.e., I would not write something like "$\phi(x,b)$ divides over $B$" unless I am assuming $b$ is taken from $B$). – Darío G Jan 30 '18 at 08:04

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