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nLab > Latest Changes: ordinal arithmetic

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2017
    • (edited May 24th 2017)

    At well-ordering theorem there had been a request for “transfinite arithmetic”. I have now made that a redirect to ordinal arithmetic, but this entry is nothing but a stub at the moment.

    In cross-linking, I realized that presently “arithmetic” redirects to number theory. Is that a good idea?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 23rd 2017

    I’m guessing transfinite arithmetic could conceivably also refer to cardinal arithmetic, where the arithmetic operations are defined quite differently from those of ordinal arithmetic (even at the finite level, the primary meaning of arithmetic seems intensionally more like a cardinal notion, not an ordinal notion, even if extensionally they wind up the same there). I’ll add that “transfinite arithmetic” is not really a term in my own lexicon.

    While you’re probably right that the intended link should go to ordinal arithmetic, another solution might be to change the phrase “transfinite arithmetic” at well-ordering theorem to “ordinal arithmetic”. But I don’t know how others would feel about that.

    It makes sense to me to have a separate page “arithmetic”, one of whose functions might be to disambiguate the various senses of arithmetic.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2017

    Thanks for the feedback. I went ahead and collected at arithmetic pointer to all related entries that we have. Then I created cardianal arithmetic with the corresponding material that was at cardinal and made transfinite arithmetic point to this and to ordinal arithmetic.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 24th 2017

    Okay, cool; thanks. I may add some prose to arithmetic later.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 24th 2017
    • (edited May 24th 2017)

    Okay, I did add some prose to arithmetic, and in so doing I added some to numbers as well, which led to some semblance of discussion of why we call some things (but not other things) “numbers”. This was just an attempt, obviously inviting of further discussion.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2017

    Thanks!

    I have added some more hyperlinks. Remarkably, we do have entries for all of the following (if maybe not very substantial ones):

    and now also

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