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nLab > Latest Changes: extended natural numbers object > history

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  1. starting an article on extended natural numbers objects

    v1, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 25th 2023

    Does this general notion of extended natural numbers object appear somewhere in the literature?

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 28th 2023

    I’m reminded of the object Martin Escardo likes using, the type of non-increasing boolean sequences. Classically this is the naturals plus a point at infinity. In topological models it’s the one-point compactification of N. But in computational settings it’s richer. In particular, the complement of N that sits inside is not the infinite point, but you cannot construct a term that’s not either a finite number of infinity. This sees to me to be interesting in other toposes, for instance: in the category of sheaves of a space, is the corresponding object the sheaf of continuous functions to the one-point compactification of N? In the way the Dedekind reals are the real-valued functions, and the Cauchy reals are the locally constant real functions.

  4. As the creator of this article from a time before the nLab got stricter about sourcing, I’m going to merge the article into the extended natural numbers. There is no need for having two articles on basically the same subject (see the “universal property” section of the extended natural numbers article). And what is unique in this article is probably not in the existing literature under this name - if anything, such objects in categories similar to R-Vec are probably called “power series objects” instead of “extended natural numbers objects”.

    diff, v4, current

  5. Just noticed Todd Trimble’s question and David Roberts’ response here. I think my comment in the edit already answers Todd Trimble’s question.

    To reply to David Roberts, Martin Escardo is working in constructive mathematics and in type theory (see e.g. TypeTopology) and the extended natural numbers he is using is already discussed in the “in constructive mathematics” section of extended natural numbers, so this is really not the right article for Martin Escardo’s work.

  6. I think there is now a proper way to delete articles? In the past I know that people used to rename the page to “old article title > history” with a pointer to the merged article.

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