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I created continuum hypothesis.
Thanks! I did some reformatting. Here are few suggestions to make better-looking and more useful pages:
\coloneqq
generally looks better than writing :=
{|A|}
rather than just |A|
''
; use double-quotation marks "
[[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.How did we not already have this article?
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…
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 ”(…)”.
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.
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?
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”.
What is the difference?
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.
Is there any possible meaning of “set of real numbers” other than “subset of a (chosen) set of all real numbers”?
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…
Oh! I think the vast majority of mathematicians would consider “a set of real numbers” to mean a set of some real numbers, like $\{10,-3,\pi,\sqrt{2}\}$.
All those evil material set theorists. But you, Brutus! :-)
@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
I was mainly linking to the blog post, not to anything in the comments. Or did I misunderstand what you’re saying?
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….
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”?
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.
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.
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”).
I would accept that if the grammar weren’t so atrocious. It makes perfect sense mathematically, but it grates on my mental ears.
There’s been a recent substantial edit by an anonymous person on CH in predicative mathematics.
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