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The intrinsic topology of a MartinLöf universe
, 2012
"... Assuming the propositional axiom of extensionality, we show that a MartinLöf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a nontrivial, decidable, ext ..."
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Assuming the propositional axiom of extensionality, we show that a MartinLöf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a nontrivial, decidable, extensional property of the universe implies the weak limited principle of omniscience. This is a theorem in type theory. Without assuming extensionality, we deduce the following metatheorem: in intensional MartinLöf type theory with a universe, there is no closed term defining a nontrivial, decidable, extensional property of the universe. 1
Constructive decidability of classical continuity
, 2012
"... We show that the following instance of the principle of excluded holds: any function on the onepoint compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof doesn’t require choice and can be understood in an ..."
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We show that the following instance of the principle of excluded holds: any function on the onepoint compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof doesn’t require choice and can be understood in any variety of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of WLPO. We use this to relate uniform continuity and searchability of the Cantor space. 1
The universe is indiscrete
, 2013
"... A construction by Hofmann and Streicher gives an interpretation of a typetheoretic universe U in any Grothendieck topos, assuming a Grothendieck universe in set theory. Voevodsky asked what space U is interpreted as in Johnstone’s topological topos. We show that its topological reflection is indiscr ..."
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A construction by Hofmann and Streicher gives an interpretation of a typetheoretic universe U in any Grothendieck topos, assuming a Grothendieck universe in set theory. Voevodsky asked what space U is interpreted as in Johnstone’s topological topos. We show that its topological reflection is indiscrete. We also offer a modelindependent, intrinsic or synthetic, description of the topology of the universe: It is a theorem in type theory that the universe is sequentially indiscrete, in the sense that any sequence of types converges to any desired type, up to equivalence. As a corollary we derive Rice’s Theorem for the universe: it cannot have any nontrivial decidable property, unless WLPO, the weak limited principle of omniscience, holds. 1