Is it permitted to define a type as containing itself in Homotopy Type Theory? I know a type of all types will lead to paradoxes but I'm unsure about the type theoretic equivelent to a Quine atom.
A type that contains itself in HOTT
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type-theory
homotopy-type-theory
1 Answers
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In HoTT every entity has a type and two types can't share the same entity.
In addition every type is a member of some $U_n$. If you had an entity "$\text{Quine} : \equiv \text{Quine} : \text{Quine}$" then Quine as a type there would have to be a $ \text{Quine} : U_n$ which would mean quine had two types.
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1I don't think this is the right resolution. If you want to make formal sense of your original question, the best way is probably to see what happens when you expand HoTT with a **new** universe $U_{\star}$ and say that $U_{\star} : U_{\star}$. ($U_{\star}$ would be in addition to the usual universes $U_n$.) You have to specify some additional things, in particular what the type of $\prod_{x : A} B_x$ is in the cases when $A$ and/or $B$ have type $U_{\star}$. – 2017-01-30