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 complex complex-geometry computable-mathematics computer-science constructive cosmology definitions 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 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 nforum 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).

nLab > Latest Changes: Eudoxus real number

Bottom of Page

1 to 8 of 8

  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 31st 2015

    Wrote an article Eudoxus real number, a concept due to Schanuel.

    • CommentRowNumber2.
    • CommentAuthorspitters
    • CommentTimeOct 31st 2015

    Thanks.

    I seem to recall that this is constructive. If so, it may be worth mentioning it.

    • CommentRowNumber3.
    • CommentAuthorJames Francese
    • CommentTimeApr 18th 2019

    I’m planning a few edits to this page but want to discuss some things here first, since I find the current page a little confusing. How can we claim that the Eudoxus reals E\mathbf{E} (Arthan’s notation, or what Street calls the “effective reals” R eff\mathbf{R}_\text{eff}) are equivalent to the Cauchy reals but not the Dedekind reals? This claim appears in the remarks after the main expo. If anything it seems like the opposite is more plausible. After all, the construction of E\mathbf{E} results in an ordered, order-complete field (hence a Dedekind-reals object), and it also seems like this construction can be totally categorified at least to a sheaf topos where an integer object is available etc. and the resulting construction is the real numbers object we all know and love, isomorphic to the sheaf of continuous E/R eff−\mathbf{E}/\mathbf{R}_\text{eff}--valued functions. In particular, this line object will not in general be isomorphic to the Cauchy reals as expressed in the internal logic, this equivalence requiring weak choice. Lastly, I understand the point that we should view the Eudoxus reals as their own intrinsic construction, not aiming to replicate either the Cauchy or Dedekind reals, but this discussion is definitely haunted by the kind of constructivist question which wants to ask..which numbers are Eudoxus reals? That question is now outdated precisely due to the idea of categorification but I’d like to make some changes along the lines suggested here: clarify both the constructive and categorical content of the Eudoxus reals.

    Also, what do we think is the weakest category in which the Eudoxian construction is possible? Is it indeed a Π\Pi-pretopos with NNO?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 18th 2019

    Schanuel describes the isomorphism between Cauchy and Eudoxus reals as

    The maps in both directions are easy: send the real r to the map ’multiply by r and round down’, and send the almost homomorphism f to the limit of the Cauchy sequence f(n)/n.

    Rounding down might cause problems for internalised constructions.

    It does indeed look like a Π\Pi-pretopos with NNO is enough to construct internal Eudoxus reals, but I didn’t go through with a fine-toothed comb.

    • CommentRowNumber5.
    • CommentAuthorvarkor
    • CommentTimeSep 22nd 2023

    Add a link to a later reference of Street.

    diff, v3, current

  6. Added Emily Riehl’s blog post on the n-Caategory Café about a construction of the Eudoxus real numbers

    George Samson

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeNov 2nd 2023

    It was pointed out by Lawrence Paulson that the motivation seems similar to that of de Bruijn’s paper Defining reals without the use of rationals. It would be nice to know how they relate (I haven’t yet had time to look at the paper).

  8. In this context, \sim should not have the preceding and following horizontal space that it should have in such expressions as a\sim b. These curly braces will mean nothing appears to the left or right of \sim and therefore that erroneous space will not be there.

    drmichaelhardy@gmail.com

    diff, v6, current

1 to 8 of 8

  1.