Remove “the empty type as a univalent universe”

]]>Yes, I agree; it doesn’t make much sense. The anonymous editor who added that didn’t see fit to say anything about it here, so I don’t know why they thought it was worthwhile, but I think it is confusing and unhelpful. If whoever added it is reading this, feel free to explain what you had in mind; otherwise I think we should remove it.

]]>Added a section regarding the definition of the empty type which uses the type of propositions.

]]>more ado about nothing:

have subdivided the sub-section “As a positive type” into three sub-sub-sections:

*Recursion principle* here: This contains essentially the old material before my edits above, expanded by commentary on interoretation of the empty elimination rule in proposition logic and in categorical semantics

*Recursion principle* here: This contains the full (dependent) inductive inference rule as above, appended – for emphasis – by amplification that this still does imply the previous recursion principle

*Eta conversion* here: This is paragraph on eta-converion by previous contributors, essentially unchanged. But this still deserves a citation.

Thinking about it, the fact that empty-recursion is still a special case of empty-induction is intuitively surprising. I understand how it readily comes about from blindly applying the same syntactic manipulations just as for any inductive type, but reading these out in words here in the case of the empty type yields a strange story, where one has to think about promoting any inhabited type to a constant family over the empty type, which, at face value, is an absurdity. I gather this works out because one really does so conditioned on the absurd assumption that there is a term in the empty type in the first place. But without the syntactic manipulations as a guide, it would be hard to see through what is going on here.

Maybe I’ll add a comment on this to the entry, tomorrow when I am more awake than now.

]]>Where the entry had a parenthetical hint that the elimination rule for the empty type seen in dependent type theory is different than previously stated, I have now added in the full dependent type inference rules: here (typeset such as to match the general pattern of the list of Examples here at *inductive type*).

Where the entry next said that

However, there is an eta-conversion rule,

I have changed that two

However, one may consider an eta-conversion rule,

because neither Coq considers this rule (as the article already mentioned) nor do the references which i have now added.

In contrast, it would be good good to add some reference where that eta-rule for the empty type is in fact considered (I don’t know any, but that does not mean much).

]]>I have touched wording, typesetting and hyperlinking of this entry, in an attempt to polish it up a little.

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