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        Questions tagged [type-theory]

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        21
        votes
        8answers
        2k views

        Good introductory book to type theory?

        I don't know anything about type theory and I would like to learn it. I'm interested to know how we can found mathematics on it. So, I would be interested by any text about type theory whose angle ...
        6
        votes
        2answers
        198 views

        When is a fold monomorphic/epimorphic

        Given a functor $F : \mathcal C \to \mathcal C$ with initial algebra $\alpha : FA \to A$, and another algebra $\xi : FX \to X$, we obtain a unique morphism $\mathsf{fold}~\xi : A \to X$ such that $\...
        5
        votes
        2answers
        598 views

        How can the simply typed lambda calculus be Turing-incomplete, yet stronger than second-order logic?

        It is well-known that the simply typed lambda calculus is strongly normalizing (for instance, Wikipedia). Hence, it is not strong enough to be Turing-complete, as also mentioned on the Wikipedia page ...
        3
        votes
        0answers
        239 views

        Formal foundations done properly [closed]

        I would want to do mathematics properly, so that the proofs of results can be trusted on instead of them being just suggestions on which results could perhaps apply. This means formulating the math in ...
        13
        votes
        1answer
        532 views

        A peculiarity of Henkin's 1950 proof of completeness for higher order logic

        My question concerns Henkin's original (1950) completeness proof https://projecteuclid.org/euclid.jsl/1183730860 for classical higher order logic and type theory relative to so-called general models. ...
        5
        votes
        1answer
        340 views

        Progress towards a computational interpretation of the univalence axiom?

        I will preface this by saying that I am not an expert on type theory. I am just a curious outsider slowly making my way through the HoTT book when I (rarely) have some spare time. I am just curious ...
        6
        votes
        1answer
        328 views

        Proof of ¬(¬1 ? ¬1) in tensorial logic

        I believe I once had a proof of this proposition, but it's been lost to the mists of time and old hard drives, so who knows if it was correct, and try as I might I can't seem to reproduce it. Is it ...
        3
        votes
        1answer
        101 views

        Internal equality for Eq-fibrations' morphisms

        I have posted this question here on M.SE but since it received little attention and since it seems difficult to find helpfule references I reposting it here. In Jacob's Categorical logic and Type ...
        11
        votes
        4answers
        1k views

        Formalizations of the idea that something is a function of something else?

        I'll state my questions upfront and attempt to motivate/explain them afterwards. Q1: Is there a direct way of expressing the relation "$y$ is a function of $x$" inside set theory? More ...
        8
        votes
        0answers
        243 views

        What metatheory proves cut elimination for Simple Type Theory?

        Gaisi Takeuti's book Proof Theory proves cut elimination for Simple Type Theory (a combined result of several researchers). Thus it proves consistency of Simple Type Theory with full comprehension in ...
        9
        votes
        0answers
        755 views

        What's the point of cubical type theory?

        I have been following through the development of homotopy type theory since 2013 because I was really interested in the foundation of mathematics. The novel idea of combining programming with homotopy ...
        2
        votes
        1answer
        180 views

        Can a type in a lower universe be formed from types in higher universes?

        A type universe is a type of small types that is closed under the basic type formation operations (dependent product, sum, coproduct etc.), that is to say for example that from $A \colon U_i$ and $x \...
        2
        votes
        0answers
        149 views

        What is the consistency limit of accumulative typing below $\omega_1^{CK}$?

        Let $Th_\zeta$ be a Mono-sorted first order theory with $\zeta$ representing some recursive ordinal notation system. Primitives: =, $\in$, $T_0, T_1, ..,T_i,..$ where i is an $\zeta$ ordinal, and ...
        2
        votes
        0answers
        64 views

        What is the relation of total functions in second order arithmetic and fast growing hierarchies?

        Answer to this questions shows that fast growing hierarchies can grow arbitrarily fast for some definition of 'arbitrary'. Can second order arithmetic define all these functions (for any ordinal) ...
        15
        votes
        2answers
        733 views

        What kind of category is generated by Cubical type theory?

        What kind of ‘category’ is Cubical type theory the internal language of? Its known that Martin-L?f type theories are the internal language of Locally cartesian closed categories, adding higher ...

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