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I am working through Toposes and Local Set Theories by Bell. I would like to know the proof of this expression from page 82:

$$(\exists ! w)\alpha, \alpha(w/true),\alpha \vdash w = true $$

I would like the proof to be based on local set theory/ Mitchell–Bénabou language (i.e., the internal language of a topos).

Any help would be much appreciated.

Andrés E. Caicedo
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Richard Southwell
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    I am not familiar with that particular book so I do not know how the logic is set up, but this should be an immediate consequence of the definition of $\exists !$. After all, the claim is, "if there exists a unique $w$ such that $\alpha$ holds, and $\alpha$ holds when $w$ is replaced with $true$, and $\alpha$ holds, then $w = true$", which is obvious. The appearance of $w$ as both a bound and free variable is unfortunate but the intent is clear... – Zhen Lin Aug 08 '20 at 10:03
  • Thanks. That makes sense classically. In which case is there redundancy ? It seems enough to say "if there exists a unique $w$ such that $\alpha$ holds, and $\alpha$ holds when $w$ is replaced with $true$ then $w=true$". Anyway, what I am really looking for is a proof based on local set theory/ Mitchell–Bénabou language. – Richard Southwell Aug 09 '20 at 08:05
  • I mean, it’s true in any topos as well. But I don’t know what rules of inference are used in that book. – Zhen Lin Aug 09 '20 at 10:26
  • That is good to know. Do you know where I might find a proof of it holding in a topos ? I am really desperate :-) Then I can solve my problem by exploiting the equivalence between topos logic and local set theories. Is $(\exists ! w)\alpha, \alpha (w / true ) \vdash w = true$ also true in any topos ? It seems I may not understand how $(\exists ! w)\alpha$ works properly. So I asked another question: https://math.stackexchange.com/questions/3784966/substitution-in-logic – Richard Southwell Aug 09 '20 at 11:14

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This may be a bit tedious so let me give you an outline (references are from Bell):

  1. Expand the definition of $\exists!$ $$\exists w.(\alpha\wedge \forall u.\alpha(w/u)\to w=u),\alpha(w/true),\alpha\vdash w=true$$

  2. Use proposition 3.7.3 to eliminate the existential on the left and introduce a fresh variable $v$, $$\alpha(w/v)\wedge\forall u.\alpha(w/u)\to v=u,\alpha(w/true),\alpha\vdash w=true$$

  3. Use proposition 3.3.1 to eliminate the conjunction on the left $$\alpha(w/v),\forall u.\alpha(w/u)\to v=u,\alpha(w/true),\alpha\vdash w=true$$

  4. Use proposition 3.4.7 to eliminate the universal on the left and substitute $u/true$ $$\alpha(w/v),\alpha(w/true)\to v=true,\alpha(w/true),\alpha\vdash w=true$$

  5. But we can also use 3.4.7 to eliminate the universal on the left and substitute $u/w$ $$\alpha(w/v),\alpha(w/w)\to v=w,\alpha(w/true),\alpha\vdash w=true$$

  6. Putting these together, (noticing that we don't even need $\alpha(w/v)$, and also that $\alpha(w/w) = \alpha$), we obtain

$$\alpha\to v = w, \alpha(w/true)\to v = true,\alpha(w/true),\alpha\vdash w = true$$

The result then follows by applying transitivity (and a few applications of modus ponens/cut).

Couchy
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  • Wow, thank you a million times. It seems a bit crazy to me that Bell just gives this as a one liner. Since you are one of the few people with the knowledge I am trying to aquire from reading Bell, it would be nice to be in touch, so if you have interest in conversing about local set theory, or collaborating feel free to drop me an email.at richardsouthwell254@gmail.com. I want to write a domain specific language based on local set theory. To this end my other big question, is what kind of local set theory axioms yield a given category of presheaves. But that is for another day. – Richard Southwell Aug 09 '20 at 22:13