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Wrote an article Eudoxus real number, a concept due to Schanuel.
Thanks.
I seem to recall that this is constructive. If so, it may be worth mentioning it.
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 $\mathbf{E}$ (Arthan’s notation, or what Street calls the “effective reals” $\mathbf{R}_\text{eff}$) 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 $\mathbf{E}$ 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 $\mathbf{E}/\mathbf{R}_\text{eff}-$-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 That question is now outdated precisely due to the idea of categorification but I’d like to make some changes along the lines suggested here: clarify both the constructive and categorical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 $\Pi$-pretopos with NNO?
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 $\Pi$-pretopos with NNO is enough to construct internal Eudoxus reals, but I didn’t go through with a fine-toothed comb.
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).
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
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