# Brief history of Cardinal Characteristics of the Continuum

Cardinal characteristics of the continuum (CCC) are cardinals which are associated with naturally arising combinatorial properties of "the continuum".

The reason that "the continuum" is qualified, is that sometimes it is better to think about it as the Cantor space, other times as the Baire space, and sometimes as $\Bbb R$ itself, or as $[0,1]$. While different, they are all Polish spaces, and therefore Borel isomorphic.

For example, we can ask what is the smallest size family of functions $\cal F\subseteq\omega^\omega$ such that for every $g\in\omega^\omega$ there is some $f\in\cal F$ such that $g\leq f$ everywhere, or at least $g\leq^f$, namely there is some $m$, such that for all $n>m$, $g(n)\leq f(n)$.

One can show that this cardinal, also known as the dominating number and usually denoted by $\frak d$, is uncountable, and of course bounded by $2^{\aleph_0}$. We can prove, for example, that its cofinality is uncountable. And it is consistently taking many different values.

Some are more topological or measure-theoretic in nature, e.g. what is the smallest cardinality of a non-meager set, or how many null sets are needed to cover the whole space.

There are many more CCCs, some more famous (e.g. $\frak p$ and $\frak t$ which made headlines over the last summer), and some are less famous (e.g. $\frak h$ the shattering number).

Where can I find a brief history of the research into CCCs?

• math.lsa.umich.edu/~ablass/hbk.pdf – Joel David Hamkins Dec 14 '17 at 11:45
• I don't see a cogent historical overview, with some description of the development of the field. Can you be more specific? (I'm sure that reading the entire chapter, and following all the references and so on would be enough for one to produce this sort of historical overview. I am asking if that already exists, I don't want to do it myself at this moment.) – Asaf Karagila Dec 14 '17 at 11:49
• (On a different note, I'd be happy to hear more about the downvote. Is it because I "didn't research enough"? Or is it because I think that a 103 pages paper without a particularly strong historical overview section is not the right way to go? Or is it something else entirely?) – Asaf Karagila Dec 14 '17 at 12:59
• Historical discussions of certain aspects of this topic can be found in Krzysztof Ciesielski's 1997 survey article Set theoretic real analysis. – Dave L Renfro Dec 14 '17 at 14:32

There is a historical note to §3 (“Six cardinals”) in chapter 3 (“The Integers and Topology”, by Erik K. van Douwen) in the 1984 Handbook of Set-Theoretical Topology edited by Kunen and Vaughan. For the sake of MathOverflow's completeness, here is an excerpt of it:

$\mathfrak{b}$, $\mathfrak{p}$ and $\mathfrak{t}$ are due to Rothberger [1939, 1948], $\mathfrak{d}$ to Katětov [1960A], $\mathfrak{a}$ to Hechler [1972b] and Solomon, and $\mathfrak{s}$ to Booth. (For simplicity we give someone credit for a cardinal $\mathfrak{k}$ even if he or she only considers the possibility that $\mathfrak{k}$ is or is not equal to $\omega_1$ or $\mathfrak{c}$. Also, “is due to” abbreviates some laborious phrase with “the earliest reference we are aware of”.) […]

There have been many notations for these cardinals. For example, $\mathfrak{b}$ is called $\aleph_\eta$ by Rothberger [1939], $K_8$ by Hechler [1972], $\lambda_3$ by Solomon and $\xi$ by Burke and van Douwen. Moreover, Rothberger [1939, 1941] uses $B(\kappa)$ for “every $F\subseteq[\omega]^\omega$ with $|F|=\kappa$ is bounded”, i.e., for $\kappa^+\leq\mathfrak{b}$ ($B$ for ‘bornée’), Tall [2000] uses $P(\kappa)$ for $\kappa\leq\mathfrak{p}$ ($P$ because Rudin's proof that $\beta\omega-\omega$ has a $P$-point under $\mathfrak{c}=\omega_1$ works if $P(\mathfrak{c})$), and van Douwen [1976A] uses $\mathrm{BF}(\kappa)$ for $\kappa\leq\mathfrak{b}$ ($\mathrm{BF}$ for ‘bounded functions’).

We here introduce yet another notation which we hope will be definitive. Our letters were chosen to be mnemonic (of course $\mathfrak{p}$ was inpsired by $P(\kappa)$, and ‘pseudo-intersection’ was created to make $\mathfrak{p}$ look mnemonic), and they are lower case German (whenever available) because $\mathfrak{c}$ is lower case German. (Note that $\mathfrak{c}$ is mnemonic: the cardinality of the continuum.) Jerry Vaughan [1979] has independently come up with this lower case German convention, and, with one exception, even chose the same letters: we now agree about the letters for the eight cardinals mentioned.

In Theorem 3.1, $\omega_1\leq\mathfrak{p}$ is due to Hausdorff [p.244], $\mathfrak{t}\leq\mathfrak{p}$ and (b) and (c) to Rothberger [1948…], $\mathfrak{b}\leq\mathfrak{a}$ to Solomon […].

The 1939 reference to Fritz Rothberger is a bit confusing because the quoted chapter lists two 1939 references, labelad 1939A and 1939B, to papers by Rothberger, the former of which was actually published in 1938:

Both are inspired by previous works by Sierpiński. The former paper considers bounding family but only the latter defines (under the name $\aleph_\eta$) what would be called $\mathfrak{b}$. So I suppose the latter is a reasonable candidate for the first occurrence of a cardinal characteristic of the continuum in the mathematical literature.

• In view of my experience with Fundamenta Mathematicae's publication schedules, I wouldn't be surprised if the claimed 1939 publication date for a paper in the 1938 volume is actually correct. I believe one of my papers in FM has an official publication date earlier than when I wrote it. – Andreas Blass Dec 14 '17 at 18:17
• Another tangential comment: You can learn a lot about cardinal characteristics by reading lots of papers by Rothberger. And you'll also learn the history pretty well, because in those days Rothberger was the history. I've been amazed by how far ahead of his time he was. – Andreas Blass Dec 14 '17 at 18:18
• @Andreas: You made me look in my FM paper to see if the dates match reality... – Asaf Karagila Dec 14 '17 at 18:32
• @Andreas Blass: On the other hand, the during much of the 1980s, papers published in FM's "sister journal" Colloquium Mathematicum typically appeared in volumes that were dated several years (6 or 7 in some cases, I believe) later than the papers' submission dates. – Dave L Renfro Dec 14 '17 at 19:33

Juris Steprans's article in the Handbook of the History of Logic: "History of the Continuum in the 20th century".

http://www.math.yorku.ca/~steprans/Research/PDFSOfArticles/hoc2INDEXED.pdf