Not signed in (Sign In)

Not signed in

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

  • Sign in using OpenID

Site Tag Cloud

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 comma 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 finite 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 k-theory lie-theory limits linear linear-algebra locale localization logic mathematics 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

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 24th 2010

    If I have a well-ordered set (J,<)(J,\lt), a countably infinite subset IJI \subset J with the induced order and a given aIa\in I, what can we say about the subset L(a):={bI|b<a}L(a) := \{b\in I| b \lt a\}? What conditions are needed on J,IJ,I or aa to say it is finite? Are there such conditions?

    Edit: Hmm, I suppose J=IJ = I could be ω+1\omega+1 and aa could be the top element, and then L(a)=ωL(a) = \omega. I’m still interested in some formal conditions, and I feel my ordinal-fu is insufficient. This seems too much like a homework question, else I’d put it on MathOverflow.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMay 24th 2010

    I can’t think of anything you could say that would ensure it other than the tautological “there are only finitely many elements preceeding aa”.

    • CommentRowNumber3.
    • CommentAuthorIan_Durham
    • CommentTimeMay 24th 2010

    Could you say something like aa is the least element of the subset of all elements greater than bb, or is that equivalent to Mike’s tautology?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 25th 2010

    @Mike

    I think you’re right. In the end I didn’t need this, and I’m sure glad I didn’t ask on MO!