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2-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 infinity integration integration-theory k-theory kan lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar 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 string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 27th 2019
   • (edited Jan 27th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of existence of linear extensions ([here](https://ncatlab.org/nlab/show/partial+order#LinearExtensionPrinciple)) Will give this its own entry at _[[linear extension of a partial order]]_, for ease of referencing <a href="https://ncatlab.org/nlab/revision/diff/partial+order/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/partial+order/50">v50</a>, <a href="https://ncatlab.org/nlab/show/partial+order">current</a>

   added statement of existence of linear extensions (here)

   Will give this its own entry at linear extension of a partial order, for ease of referencing

   diff, v50, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 27th 2019
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexFor what interest this may have, I'd imagine this proposition would be provable using just the [[ultrafilter theorem]] in place of the axiom of choice. (But I'd have to write out a proof to be sure.)

   For what interest this may have, I’d imagine this proposition would be provable using just the ultrafilter theorem in place of the axiom of choice. (But I’d have to write out a proof to be sure.)