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Created Dedekind completion. Probably not very satisfactory, but I lifted the main definition from Paul Taylor’s page on Dedekind cuts, so should be ok with a little tweaking.
Interesting, I have a question brewing from Jamie’s paper on Dedekind completion as it is one of the criteria to obtain complex numbers from a $\dagger$-category. I was curious what number system the $\dagger$-category would represent if you did not enforce Dedekind completeness. This is the point where it seems the continuum is introduced and as Jamie admits, is the one criteria that is questionable on purely physical grounds.
The section Interval cuts at Dedekind cut may interest you in this regard, in that it is posed that interval cuts represent ’real world measurements’. It would be interesting to see how the details of Jamie’s paper would go if an approach like that was used, instead of Dedekind cuts (i.e. real or complex numbers)
Oh! I’ll have a look. Thanks. I also added a link to Paul Taylor’s page to Dedekind completeness
I put in a couple of redirects and some links.
I just broke the link in comment #3 to interval cuts by giving that section a name; but now my link should never break even if the article is later rearranged.
I deleted the ’sanity check’ comment, and added a link to the when referring to Dedekind completion as a universal construction. I’m also adding a comment echoing the last paragraph of Dedekind cut, that one can use other linear orders than $\mathbb{Q}$ to recover $\mathbb{R}$ as the Dedekind completion. The way I’ve said it probably needs expanding, but so does the whole page :) There’s obviously a lot more that could be said.
At some point we were going to do more work on completion in general. If I recall correctly, Dedekind completion is a variety of MacNeille completion. Todd had some fascinating thoughts on this.
The page Dedeking completion didn’t say right away what a Dedekind completion actually is. I have added a first sentence:
The Dedekind completion of a linear order is a new linear order that contains suprema for all inhabited bounded subsets.
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