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Properties of the Lattice of Theories


Joshua Wrigley
(@jwrigley)
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My purpose in starting this thread is to garner what is understood about the lattice of theories \mathfrak{Th}_\Sigma^{\mathbb{T}}, for a given geometric theory \mathbb{T}, that is the theories \mathbb{T}' \supseteq \mathbb{T} with the order defined by:

 \mathbb{T}'' \leqslant \mathbb{T}' \iff \mathbb{T}'' \vDash \varphi \vdash_{\vec{x}} \psi, \ \forall \, \varphi \vdash_{\vec{x}} \psi \in \mathbb{T}' .

I recently came across this in Olivia's Theories, Sites and Toposes (TST) where it is shown that \mathfrak{Th}_\Sigma^{\mathbb{T}} is a complete Heyting algebra.  I'm wondering what other properties we are able to deduce.

I've attached some notes giving a sufficient condition for the open subtopoi to be complemented in terms of their corresponding theories, namely the theories of the form \mathbb{T} \cup \{\, \top \vdash_{[]} \varphi\,\} are complemented in  \mathfrak{Th}_\Sigma^{\mathbb{T}} if there exists a sentence  \psi such that  \mathbb{T} \vDash \top \vdash_{[]} \varphi \lor \psi, \ \varphi \land \psi \vdash_{[]} \bot (i.e.  \varphi is complemented).

As always, let me know if there are any mistakes.  I'd like to know what else we can say about \mathfrak{Th}_\Sigma^{\mathbb{T}} (and generally generate discussion).  Of course, just reading more TST would be sound advice as I've yet to finish it.

 


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Joshua Wrigley
(@jwrigley)
Active Member
Joined: 4 months ago
Posts: 10
Topic starter  

Indeed, had I read further in TST, I would have come across the very result I proved in the attachment above in Subsection 4.2.2.4, therein described as "not hard to prove using logical arguments".  Hopefully my proof is testament to that.


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