Re: [HoTT] Re: Where is the problem with initiality?

["András Kovács" <"putta..."@gmail.com>]

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>
> I like this question. If the syntax is supposed to be the initial model,
> then it can't be a set.


Why? The syntax being a set doesn't imply that models also need to be sets.
As I understand, the goal of higher models for TT is to get rid of
set-truncation but still have a syntax which is a set.

But, then, this seems to be a statement about *extrinsic* syntax.  If,
> as Thorsten advocates, we somehow manage to produce a highly
> structured, internal description of the syntax of type theory,
> then typechecking for this syntax is, by definition, unnecessary!
>

Type checking must be performed in any case, at some level. If we use
intrinsic embedded syntaxes, then the implementation of the metalanguage
(in a meta-metalanguage) does the type checking. Hence, if we intend to
formalize a syntax of type theory which is intended as a metatheoretical
setting for mathematics, then I think decidability is quite important.


2018-06-01 21:55 GMT+02:00 Martín Hötzel Escardó <escardo...@gmail.com>:

>
>
> On Friday, 1 June 2018 18:45:42 UTC+1, Eric Finster wrote:
>>
>> I guess my question is pretty simple: why should we insist,
>> as Thorsten seems to, that the "intrinsic" syntax of type
>> theory form a set?
>>
>
> I like this question. If the syntax is supposed to be the initial model,
> then it can't be a set.
>
>  Martin
>
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