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    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2012

    Thanks! I did some reformatting. Here are few suggestions to make better-looking and more useful pages:

    • The command \coloneqq generally looks better than writing :=
    • For reasons that I don’t understand, when enclosing an object in vertical bars (e.g. to denote cardinality, absolute value, etc.) it looks better to write {|A|} rather than just |A|
    • Don’t use LaTeX-quotes “ and ''; use double-quotation marks "
    • If you make a [[link]] to a page that you think has a reasonable chance of existing already, but it comes out as a gray link, try searching the nLab for variations on the page name; there’s a good chance it exists already but someone named it slightly differently. E.g. you linked to “natural-numbers object” but the page that exists is called “natural numbers object”. Once you find the page, you can either add a redirect to it so that your alternative name works, or just update your link so that it points to the actual page. You may also find a page that is so closely related that it makes sense to redirect or link to that page instead, e.g. I redirected your link to directed diagram so that it ends up at direction.
    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJun 19th 2012

    How did we not already have this article?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2012
    I was wondering that too. *shrug*
    • CommentRowNumber5.
    • CommentAuthorStephan A Spahn
    • CommentTimeJun 20th 2012
    • (edited Jun 20th 2012)

    I did some reformatting. Here are few suggestions to make better-looking and more useful pages:

    Thanks. In principle I know these things - except {|A|}.

    In case of the natural numbers object I would prefer the spelling “natural-numbers object” since the object is not a natural object which is also a numbers object…

  1. How did we not already have this article?

    By the way, the section of forcing intended to describe forcing in terms of sheafification consists only of ”(…)”.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2012
    • (edited Jun 20th 2012)

    I have added some links. Notice that we have an entry continuum.

    By the way, the Idea-sentence seems strange where it says:

    […] every set of real numbers is either countable or has the same cardinality as all the reals

    Somehow this uses “reals” in two different senses without explaining what the point is.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJun 20th 2012

    Somehow this uses “reals” in two different senses without explaining what the point is.

    What are the two senses? I changed it to read “… as the set of all real numbers”, does that make you happier?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2012

    I think what would make me happy is if instead of “every set of real numbers” it would say “every subset of the real numbers”.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 20th 2012

    What is the difference?

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 21st 2012

    Might it be that we want the sets of real numbers to be subobjects of a given real numbers object? This is what the mathematical statement says.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2012

    Is there any possible meaning of “set of real numbers” other than “subset of a (chosen) set of all real numbers”?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2012

    Well, I misunderstood it initially as meaning a set of all real numbers, as witnessed by this discussion. But if nobody else has this problem…

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2012

    Oh! I think the vast majority of mathematicians would consider “a set of real numbers” to mean a set of some real numbers, like {10,3,π,2}\{10,-3,\pi,\sqrt{2}\}.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2012

    All those evil material set theorists. But you, Brutus! :-)

    • CommentRowNumber16.
    • CommentAuthorMirco Richter
    • CommentTimeJun 21st 2012
    @Urs: Can you explain the term: "material set theorists' ? I think it is the third time I read that from you and I would like to know the idea behind that 'wordplay' ...

    If I should make a guess it is because if we forget the (higher layers of) morphisms of an $(\infty,1)$-category (and the morphism part of their functors) what is left is set theory with functions and then you regard the object part of a $(\infty,1)-category as its material part?

    (Don't takes this question too serious. Just kind of interesting ...)
    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2012

    @Urs: I don’t think it has anything to do with material-ness. Even structurally, the set of all real numbers is defined uniquely up to unique isomorphism, so I can talk about “real numbers” without needing to be explicit about “which set-of-all-real-numbers” I’m talking about.

    @Mirco: http://golem.ph.utexas.edu/category/2009/12/syntax_semantics_and_structura_1.html

    • CommentRowNumber18.
    • CommentAuthorMirco Richter
    • CommentTimeJun 21st 2012
    • (edited Jun 22nd 2012)
    Damn, someone should define some 'tree overlays' to all those nice nCat posts of the past. Its hard to find a good entry point to most of your stuff because it is so 'net-like'. In this case a 'evolution_of_homotopy_type_theory_in_the_nCafe' tree overlay to all those linked articles with a ROOT would be nice.

    Of course you can argue that different people would like to start at different points, but I think having at least some recommended roots would be a good way to get more people into the nRevolution....

    And by the way, I think this is a good idea for the nLab, too...
    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2012

    I was mainly linking to the blog post, not to anything in the comments. Or did I misunderstand what you’re saying?

    • CommentRowNumber20.
    • CommentAuthorMirco Richter
    • CommentTimeJun 22nd 2012
    • (edited Jun 22nd 2012)
    I think you get me wrong here. Your link was fine as an answer to my question on Urs. My last post was just an impression. Reading your linked post made me think immediately that it is not the right place to start if I want to fully understandwhat was written there. But that is of course my problem...
    • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2012

    Ah, I see what you mean. Yes, it’s tough; I’ve had that sort of problem trying to read old Cafe posts from before I started following and posting myself. And, for that matter, reading some math papers! (I particularly have that problem reading papers written by computer scientists.) That’s one reason we try to provide links to “previous posts on this subject”, of course. And you can always ask questions….

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2012
    • (edited Jun 22nd 2012)

    Mike,

    structurally, I cannot look at a set and say “Oh, look, it is ’a set of real numbers’!”. I can only ask if it is a subset of the real numbers.

    Also, right after the Idea-section is over, the entry continuum hypothesis does adopt this perspective.

    Just let me know, is there reason to complain if I go ahead and edit the entry to make “every set of real numbers” become “every subset of the real numbers”?

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2012

    structurally, I cannot look at a set and say “Oh, look, it is ’a set of real numbers’!”. I can only ask if it is a subset of the real numbers.

    Actually, you can’t do either of those things with a “set” all by itself. Structurally, “being a subset of the real numbers” is not a property of a set, it is structure on a set (namely, a monomorphism to another set, which in turn is equipped with structure making it a set-of-all-the-real-numbers). And if you have that structure, you can just as well say “it is a set of real numbers” as “it is a subset of the real numbers”. As far as I can see, this is not a “different perspective”; it is just two phrases that mean exactly the same thing. I think we should feel free to use either one interchangeably.

    • CommentRowNumber24.
    • CommentAuthorTobyBartels
    • CommentTimeJun 27th 2012

    Grammatically, ‘a set of real numbers’ is correct but ‘a subset of the real numbers’ is not; it should be ‘a subset of the set of [all] real numbers’ instead. So ‘a set of real numbers’ is shorter; but then, ‘a subset of \mathbb{R}’ is also short.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeJun 27th 2012

    I think saying “the real numbers” to mean “the set of [all] real numbers” is one of those “abuses” of language/notation that are so common in mathematics that I would elevate them to the status of an overloaded definition (free of the perjorative connotations of the word “abuse”).

    • CommentRowNumber26.
    • CommentAuthorTobyBartels
    • CommentTimeJun 27th 2012

    I would accept that if the grammar weren’t so atrocious. It makes perfect sense mathematically, but it grates on my mental ears.

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 9th 2021

    There’s been a recent substantial edit by an anonymous person on CH in predicative mathematics.