1

Under ZFC, the real numbers can be well-ordered. So, there is some ordinal number whose cardinality is that of the continuum. Is there a standard notation for this number?

For example, the first infinite ordinal is usually denoted $\omega$, and the first uncountable ordinal is usually denoted $\omega_1$. But unless we appeal the continuum hypothesis (or something just as presumptuous), $\omega_1$ may not have cardinality of the continuum.

Here is a related question, which is just my curiosity at work: Do we know whether there is such thing as a least and/or greatest ordinal with cardinality of the continuum?

Thanks!

P.S. One last recreational-math question: Are there any other well-known measures of a number's size besides ordinals and cardinals?

MJD
  • 67,568
  • 43
  • 308
  • 617
Ben W
  • 5,336
  • Please don't ask unrelated questions on the same post. – Asaf Karagila Oct 01 '18 at 23:34
  • (And before you run to post it on a separate question, please clarify what you mean. Ordinals only measure order types, specifically of well-ordered sets. They require some structure with constraints. You can talk about arbitrary order types, but it still requires some order to it. If you start adding requirements, requiring a set is a Borel set on $[0,1]$ gives you a good measure of it, etc.) – Asaf Karagila Oct 01 '18 at 23:36
  • I admit, it is difficult to explain what I'm asking in my postscript. Perhaps a more specific example would help illustrate what I'm after. Let's take the set of naturals, and the set of even naturals. Most laypersons would assume that the set of naturals is in some sense "larger" than the set of even naturals. Mathematicians might be tempted to counter that this is false, as the naturals and the evens have identical cardinality. But frankly, I'm not altogether certain that the laypersons are incorrect here. Are cardinals really the only way to measure the size of a set? – Ben W Oct 01 '18 at 23:45
  • Ben, I've written about this before: https://math.stackexchange.com/questions/40309/cardinality-density/ https://math.stackexchange.com/questions/125412/comparing-the-sizes-of-countable-infinite-sets https://math.stackexchange.com/questions/168258/is-there-a-way-to-define-the-size-of-an-infinite-set-that-takes-into-account https://math.stackexchange.com/questions/242057/why-the-principle-of-counting-does-not-match-with-our-common-sense I hope these satisfy you. If not, your "P.S." question should really be removed and expanded into a new question on its own. – Asaf Karagila Oct 01 '18 at 23:48
  • Yes, I had thought about natural density, but it is highly specific to subsets of naturals. I'm not aware of general measures of density for sets of numbers---and yes, I realize that last sentence is a bit garbled, but hopefully you take my meaning. – Ben W Oct 02 '18 at 00:00
  • What are numbers? Real numbers? Natural numbers? Rational numbers? Ordinal numbers? Surreal numbers? Hyperreal numbers? This is exactly what I said. If you start allowing specific constraints, you might have better ways to come up with answers (again, the Lebesgue measure on Borel sets is a good example). But you keep being vague. If you find those answers as unsatisfactory, you should post a separate question where you explain why those things are not answering your actual question. And elaborate on your actual question, too. – Asaf Karagila Oct 02 '18 at 00:03
  • I appreciate your comments. They are not unsatisfactory except in a very literal sense that one can probably never be satisfied by one's inquiries. Thank you. – Ben W Oct 02 '18 at 00:08
  • Why is this [tag:recreational-mathematics]? – Peter Taylor Oct 03 '18 at 09:52

1 Answers1

2

Not really. If one assumes the axiom of choice, as one usually does, then one has that $2^{\aleph_0}$ is in fact an ordinal, an initial ordinal. So writing $\alpha<2^{\aleph_0}$ has a very clear meaning. To that end, using $\frak c$ is also quite common as denoting the cardinality of the continuum. So it is not unusual to see $\alpha<\frak c$ in set theoretical papers.

So yes, there is such a least ordinal, this is the initial ordinal (by the definition of an initial ordinal). There is no maximal ordinal, just like there is no "largest countable ordinal". If $X$ is an infinite set, there is no largest ordinal equipotent to $X$. In fact, there are exactly $|X|^+$ ordinals which are equipotent to $X$. Just like there are $\aleph_1$ countably infinite ordinals, and $\aleph_2$ ordinals of size $\aleph_1$, and so on.

Asaf Karagila
  • 405,794
  • So, if I understand you correctly, in the literature, c doubles as a label for some unspecified order type of the reals, yes? – Ben W Oct 01 '18 at 23:41
  • It is the cardinality of the reals. If cardinality also means the suitable initial ordinals, yes. But $\frak c$ is also used sometimes in contexts where the reals cannot be well-ordered, which is why I am somewhat reluctant to say "yes" outright. But again, if you were to write $\alpha<\frak c$, I think that most readers should be able to infer what you meant. – Asaf Karagila Oct 01 '18 at 23:42
  • Not an unspecified order type, but specifically the least ordinal which can be mapped bijectively with the reals. – Ned Oct 01 '18 at 23:45
  • So, there is such a least ordinal? – Ben W Oct 01 '18 at 23:46
  • @Ben: I've added a bit on that. I missed that part of your question. Sorry! – Asaf Karagila Oct 01 '18 at 23:49
  • If you want a symbol for the order type of the reals, sometimes $\lambda$ is used (and $\eta$ is used for the order type of the rationals). But it is not universally used, so you should probably remind readers of the convention if you use it. – Andrés E. Caicedo Oct 01 '18 at 23:51
  • @Andrés: And remind the readers that it is not an ordinal. – Asaf Karagila Oct 01 '18 at 23:52
  • There was a guy who once, upon being told that the order of the reals is not a well-order, asked me to help him prove that, instead, it is a projective well-order. – Andrés E. Caicedo Oct 01 '18 at 23:54
  • @Asaf: I thought order types are identified with ordinals, sorta like, say, a.e.-equivalent functions are identified in function spaces. – Ben W Oct 01 '18 at 23:57
  • @Ben: Order types are just a way to think about an ordered set without paying much attention to the specific underlying set. The ordinals simply provide us with canonical underlying sets for well-ordered sets. Since everything is so... canonical and unique with ordinals, that can sometimes even be a bit helpful to pay attention to the elements of ordinals. But generally, not that much. – Asaf Karagila Oct 01 '18 at 23:59