Let $L$ be a given language and $\mathcal{T}$ be the set of complete theories in that language. We give a topology to $\mathcal{T}$ by considering as basic open sets the sets of the form $\langle \phi \rangle = \{T \in \mathcal{T} \; | \; T \models \phi\}$. Its not difficult to see that this topology is Hausdorff and totally disconnected, and moreover the compactness theorem guarantees that it also compact.

Now, I'm trying to understand what it means for a set of theories to converge to a given theory in this space. If the set is countable, I thought of the following definition: let $T_0, T_1, T_2, \dots$ be a countable sequence of theories in $\mathcal{T}$ and $T$ a given theory. For each open set $U$, associate $Z_U = \{ n \in \mathbb{N} \; | \; T_n \in U\}$. Let $F = \{ U \; | \; Z_U \text{ is cofinite}\}$. Then $F$ is a filter and we say that $T_0, T_1, \dots$ converge to $T$ iff every neighborhood of $T$ is contained in $F$. Intuitively, this seems to mean that for every set of formulas validated by $T$, there are infinitely many theories in the sequence that also validate it, so these theories are all "close" to $T$.

Is this definition of convergence right? Can it be generalized for sets of theories of higher cardinalities, i.e. can we form for a given cardinal $\aleph_\alpha$ the set $F = \{U \; | \; |Z_U| = \aleph_\alpha\}$ (modifying the definition of $Z_U$) and then give the same definition (I mean, does this definition work)? Moreover, is my intuition correct, or is there a more intuitive way of thinking about this?

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    $\begingroup$ What you've written is exactly the usual definition of convergence in a topological space, and it's also a reasonable notion of convergence for theories. $\endgroup$ – Alex Kruckman Sep 23 '18 at 13:24
  • $\begingroup$ @AlexKruckman - Including for higher cardinalities? $\endgroup$ – Nagase Sep 23 '18 at 13:25
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    $\begingroup$ Well, usually in a topological space, you define convergence for sequences (indexed by the natural numbers). But there are more general notions of convergence, one of which is obtained by looking at more general filters than the coconuts filter. This seems to be what you're going for. $\endgroup$ – Alex Kruckman Sep 23 '18 at 13:30

In any topological space $X$, we have the following standard definition: A sequence of points $(x_i)_{i\in \mathbb{N}}$ converges to a point $x$ if and only if for every open neighborhood $U$ of $x$, all but finitely many of the $x_i$ are in $U$ (equivalently, $\{i\in \mathbb{N}\mid x_i\in U\}\in F$, where $F$ is the cofinite filter on $\mathbb{N}$). [Note that my notation is slightly different than yours. Your $F$ is the pushforward of the cofinite filter on $\mathbb{N}$ by the indexing map $g\colon i\mapsto x_i$. That is, a set $U$ is in your $F$ if and only if $g^{-1}(U) = \{i\in \mathbb{N}\mid x_i\in U\} = Z_U$ is in my $F$.]

When you use this definition in the context of the Stone space of complete $L$-theories, you get a reasonable notion of the convergence of a sequence of theories $(T_i)_{i\in \mathbb{N}}$ to a limit theory $T$. This notion of convergence agrees with the intuitive idea that ultraproducts are a kind of "logical limit" of structures. Namely, $(T_i)_{i\in \mathbb{N}}$ converges to $T$, and if $M_i\models T_i$ for all $i\in \mathbb{N}$, then for any non-principal ultrafilter $U$ on $\mathbb{N}$, we have $\prod_{i\in I} M_i / U\models T$ by Łoś's theorem (this is because every non-principal ultrafilter contains the cofinite filter).

You ask about generalizing this to sets of theories of higher cardinality. You can do that, and you exactly get the theory of convergence in general topology using filters instead of sequences. But your proposed definition might be problematic: if your set of points has size $\aleph_\alpha$, the set $F = \{U \mid |Z_U| = \aleph_\alpha\}$ might not be a filter! Indeed, you can split a set of size $\aleph_\alpha$ into two disjoint sets both of size $\aleph_\alpha$ (when $\alpha = 0$, just take the even and odd numbers). You can still make the definition, but if you don't use a filter, you might get multiple limit points. If the index set is $I$, with $|I| = \aleph_\alpha$, then the proper generalization of the cofinite filter is $\{U\mid |I\setminus Z_U|<\aleph_\alpha\}$.

You can then easily generalize the connection with ultraproducts above. But now if you fix a filter $F$ on the index set $I$, and use the pushforward filter to define a notion of convergence in the space of theories, you'll find that $\prod_{i\in I} M_i/U\models T$ exactly when the ultrafilter $U$ extends the filter $F$.

  • $\begingroup$ That's really nice! Thanks a lot! $\endgroup$ – Nagase Sep 23 '18 at 16:21

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