Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeDec 30th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCreated [[double-negation shift]], with a proof that it is equivalent to double-negated [[excluded middle]].

   Created double-negation shift, with a proof that it is equivalent to double-negated excluded middle.

2. • CommentRowNumber2.
   • CommentAuthorJonasFrey
   • CommentTimeDec 31st 2017
   • (edited Dec 31st 2017)
   • PermaLink
   Author: JonasFrey
   Format: MarkdownItexIf I'm not mistaken, this principle holds in the topos of sheaves on a space (or locale) $X$ iff $X$ contains a *dense locally decidable open*, i.e. a dense open set $U\in O(X)$ such that for all $V\subseteq U$ we have $U\subseteq V\cup\neg V$. An example is the Sierpinski space, a counterexample is the real line. (the negation is the closure of the complement)

   If I’m not mistaken, this principle holds in the topos of sheaves on a space (or locale) $X$ iff $X$ contains a dense locally decidable open, i.e. a dense open set $U\in O(X)$ such that for all $V\subseteq U$ we have $U\subseteq V\cup\neg V$. An example is the Sierpinski space, a counterexample is the real line. (the negation is the closure of the complement)

3. • CommentRowNumber3.
   • CommentAuthorJonasFrey
   • CommentTimeDec 31st 2017
   • (edited Dec 31st 2017)
   • PermaLink
   Author: JonasFrey
   Format: MarkdownItexI suppose at least in the spatial case the local decidability is equivalent to discreteness, so the condition means that $X$ has a *dense and discrete open subspace*. An example satisfying T_2 is the 1-point compacitification of an infinite set.

   I suppose at least in the spatial case the local decidability is equivalent to discreteness, so the condition means that $X$ has a dense and discrete open subspace.

   An example satisfying T_2 is the 1-point compacitification of an infinite set.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 31st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIf you'd like to add this to the page with a proof, feel free!

   If you’d like to add this to the page with a proof, feel free!

5. • CommentRowNumber5.
   • CommentAuthorJonasFrey
   • CommentTimeJan 2nd 2018
   • PermaLink
   Author: JonasFrey
   Format: MarkdownItexDid it, at least with proof outlines. I hope I didn't make any mistakes!

   Did it, at least with proof outlines. I hope I didn’t make any mistakes!

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJan 2nd 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThanks!

   Thanks!

7. • CommentRowNumber7.
   • CommentAuthorspitters
   • CommentTimeJul 17th 2018
   • PermaLink
   Author: spitters
   Format: MarkdownItexDNS holds in every Kripke model with finite frame. <a href="https://ncatlab.org/nlab/revision/diff/double-negation+shift/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/double-negation+shift/5">v5</a>, <a href="https://ncatlab.org/nlab/show/double-negation+shift">current</a>

   DNS holds in every Kripke model with finite frame.

   diff, v5, current