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If I'm not mistaken, the space $\mathcal{T}$ of all complete theories of a countable language is a compact Hausdorff space, and moreover it is second-countable, since it has as its base the sets of the form $\langle \phi \rangle = \{T \in \mathcal{T} \; | \; T \models \phi\}$, which is countable provided that the language is countable. If I'm not mistaken (and I may very well be!), any space that satisfies these three conditions is metrizable.

Question: Is there any interesting metric on this space?

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  • $\begingroup$ You're right that $\mathcal{T}$ is metrizable, and there are many metrics on it which are compatible with the topology. For you, what would quality any particular metric as interesting? $\endgroup$ – Alex Kruckman Sep 23 '18 at 15:45
  • $\begingroup$ @AlexKruckman - I'm not sure I can make it much more precise. I was thinking about a metric that is enlightening in the case of an important theorem or that helped in understanding an important property of this space. Maybe some type of metric that connected with syntactic criteria (e.g. quantifier rank of a formula that corresponded to a basic open)? If you could point me to some references, that would also be helpful. $\endgroup$ – Nagase Sep 23 '18 at 16:25
  • $\begingroup$ The standard kind of metric looks like the following: Enumerate the sentences in the language as $(\varphi_n)_{n\in \mathbb{N}}$. Then let $d(T,T') = \sum_{n\in \mathbb{N}} \delta_n$, where $\delta_n = 2^{-n}$ if $T$ and $T'$ disagree about $\varphi_n$, and $\delta_n = 0$ otherwise. Of course, different enumerations of the sentences give different metrics. Metrizability of Stone spaces (for countable languages) is sometimes useful, but I'm not aware of any situation where the choice of metric matters. $\endgroup$ – Alex Kruckman Sep 23 '18 at 16:47
  • $\begingroup$ @AlexKruckman - is there any relation between this metric and the complete binary tree? I vaguely remember something to the effect that complete theories could be thought as infinite branches and that it was possible to measure their distance using the nodes where they diverged. $\endgroup$ – Nagase Sep 23 '18 at 17:36
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    $\begingroup$ An enumeration of the sentences identifies a theory with a binary sequence, i.e. an path through the complete binary tree. This identifies the space of all theories with a subspace of the space $2^\omega$ of all such paths, and the metric I described above is the restriction of the standard metric on $2^\omega$. @Nagase $\endgroup$ – Alex Kruckman Sep 23 '18 at 21:02

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