https://math.codidact.com/categories/41/tags/3924.rssNew Posts Tagged 'type-theory' - MathematicsMathematics - Codidact2023-06-16T22:04:25Zhttps://math.codidact.com/posts/288016Considering the "type" $\Pi x:A.\Pi y:B(x).Type$user205https://math.codidact.com/users/592452023-04-01T22:21:59Z2023-06-16T22:04:25Z<p>I'm still trying to wrap my head around introducing type theories by means of LF. Suppose I declare the following constants:</p>
<p>$A:Type, \\ B:(El(A))Type$ and the necessary constants for th...https://math.codidact.com/posts/288007Defining the two universes from Calculus of Constructions in LFuser205https://math.codidact.com/users/592452023-03-30T21:46:23Z2023-03-30T22:08:56Z<p>Consider LF, the logical framework used to define UTT (unified theory of dependent types). The next two quotes are from <a href="https://global.oup.com/academic/product/computation-and-reasonin...https://math.codidact.com/posts/287995On Tarski-style universes in type theoryuser205https://math.codidact.com/users/592452023-03-27T03:09:09Z2023-03-30T10:09:01Z<p>I'm looking at <a href="https://www.cs.rhul.ac.uk/home/zhaohui/universes.pdf">this note</a> about universes in type theory.</p>
<p>For Tarski-style universes (page 2), as far as I understand, t...https://math.codidact.com/posts/287991The purpose of logical frameworks in specifying type theoriesuser205https://math.codidact.com/users/592452023-03-26T14:49:03Z2023-03-30T09:32:36Z<p>I'm trying to understand the notion of a logical framework and how/why/when it's used to define type theories. I'm looking at Luo's "Computation and Reasoning" (1994), where he considers "LF, a ...https://math.codidact.com/posts/278280Can we constructively find a third element of a set $X$ satisfying $X \cong X \times X$ given two distinct elements?Derek Elkinshttps://math.codidact.com/users/533982020-09-29T01:50:02Z2020-09-29T01:50:02Z<p>Consider a set $X$ such that $X \cong X\times X$. It's quite easy to prove in a classical set theory, e.g. <strong>ZF</strong>, that $X$ must be the empty set or a singleton set or an infinite s...