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Im a beginner in (infinite) set theory, So keep that in mind. I know cantor's diagonal, cantors pairing function , the ZFC axioms and basic stuff like that.

Im very intrested in infinite sets and in particular cardinality and the continuüm hypothesis.

Now I read an article ( see link ) where They mention that - If I understand correctly - that two types of cardinality are equal : card(p) = card(t).

Im not sure If $p$ and $t$ are the standard notation for these sets / Cardinals / ordinals.

But the article says who worked on it , So I hope you guys can explain $p$ and $t$ to a beginner like me.

It is Said - I think - that $p$ and $t$ are candidates to break CH.

In case the article goes away here is a quote :

INFINITY

Mathematicians Measure Infinities and Find They’re Equal By KEVIN HARTNETT September 12, 2017

Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories. 70

Colors Collective for Quanta Magazine In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers.

The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University.

In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal, much to the surprise of mathematicians.

“It was certainly my opinion, and the general opinion, that p should be less than t,” Shelah said.

Here is the article

https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/

So What is this all about ?

I looked on arxiv but I was not sure If there was a connection.

Also quantamagazine says it is a big result , but I have not heard anyone mention it ??

Now that $p = t$ I wonder If They are both countable or both uncountable ... or still candidates to break CH ??

mick
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  • I have no idea what the specific cardinalities are, but I am fairly certain you should disregard most of what is written in that article. Especially the connection to CH seems like it is unlikely to be relevant, seeing as CH is known to be independent of ZFC, so certainly these cardinalities will not be able to "break" CH. – Tobias Kildetoft Sep 18 '17 at 08:59
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    This is answered in the answers to Is there a bijection.... – Mark S. Sep 18 '17 at 09:04
  • The connection with CH is this - $\mathfrak{p}$ and $\mathfrak{t}$ are both examples of cardinal characteristics of the continuum (see my answer to the linked question above), which in particular are (i) necessarily uncountable and of size at most continuum, but (ii) consistently strictly less than the continuum. Saying that they are "candidates to break CH" is missing the point ... what's true is that they are amongst the "naturally definable" infinite cardinalities which could be strictly between $\aleph_0$ and $2^{\aleph_0}$. But this really isn't rare, e.g. $\aleph_1$ is such a cardinal! – Noah Schweber Sep 20 '17 at 18:21
  • @TobiasKildetoft I think the linked article is mostly good, actually (at least for popsci). In particular, this passage is exactly right: "Both sets are larger than the natural numbers [and] p is always less than or equal to t. Therefore, if p is less than t, then . . . the continuum hypothesis would be false. Mathematicians tended to assume that the relationship between p and t couldn’t be proved within the framework of set theory, but they couldn’t establish the independence of the problem either." That said, the end is weird: (cont'd) – Noah Schweber Sep 20 '17 at 18:25
  • the line "Their work also finally puts to rest a problem that mathematicians had hoped would help settle the continuum hypothesis." is bizarre to me. The alternative to Malliaris/Shelah would be that $\mathfrak{p}<\mathfrak{t}$ is consistent, but we only get an argument against CH from an argument that $\mathfrak{p}<\mathfrak{t}$ (or similar) ought to be adopted as an axiom; conversely, MS only argues for CH insofar as one believes that every uncountable cardinal is either $\mathfrak{p}$ or $\mathfrak{t}$, which nobody does. So it ends on a bad note. But overall, it seems fine to me. – Noah Schweber Sep 20 '17 at 18:27
  • @NoahSchweber I suppose you are right that as popsci goes, it is pretty good. I just find the way the refer to some of these things misleading (and the questions here seem to confirm that). They start out with the very imprecise and potentially misleading statements, then only later do they make things more precise and correct. – Tobias Kildetoft Sep 20 '17 at 18:30
  • To the OP: I'm a bit confused by your remark that "Also quantamagazine says it is a big result , but I have not heard anyone mention it ??" First of all, quanta mentioned it. More seriously, I think you're looking in the wrong places: the importance of a mathematical result isn't determined by how loudly it's broadcast in the popular literature. Within the mathematical community, one good indication of the significance of $\mathfrak{p}=\mathfrak{t}$ is this PNAS article. – Noah Schweber Sep 20 '17 at 18:30
  • One unfortunate feature of how math is communicated is that it can be extremely hard to tell when a result is particularly important "from the outside," but that's the way it is; having a popsci article generally is good evidence that a result is important, but not having a popsci article, or many popsci articles, is no evidence of anything. (Incidentally, you mention having trouble finding it in arxiv; the original paper is this one.) – Noah Schweber Sep 20 '17 at 18:35
  • @NoahSchweber That is actually slightly confusing, since that paper also seems to have been published (in JAMS). So which of the non-arXiv papers if the one that proved the result? Or is the PNAS paper a sort of summary thing (I am not really familiar with PNAS)? – Tobias Kildetoft Sep 20 '17 at 18:39
  • @TobiasKildetoft The PNAS paper is a summary paper. – Noah Schweber Sep 20 '17 at 18:39
  • @NoahSchweber I see, thanks. That also explains why the PNAS paper does not have a regular MathSciNet review but just a quote from the papers own summary. – Tobias Kildetoft Sep 20 '17 at 18:46
  • @NoahSchweber so presumably neither of these can measure the size of a set? – it's a hire car baby Nov 12 '17 at 20:15
  • @RobertFrost I don't know what you mean. They're cardinals; they measure the size of sets the same way that any other cardinal does. – Noah Schweber Nov 12 '17 at 20:17
  • @NoahSchweber the bit where you said "uncountable"... and "consistently strictly less than continuum" suggested to me something else must disqualify them from being counterexamples to CH. – it's a hire car baby Nov 12 '17 at 20:32
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    @RobertFrost CCCs can be counterexamples to CH - it's just that they aren't provably strictly less than the continuum, only consistently less than the continuum. Similarly, it is consistent that $\omega_1$ is a counterexample to CH, it's just not provable. – Noah Schweber Nov 12 '17 at 20:38
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    Note that it's even possible for there to be a definable set of reals of size in between $\aleph_0$ and continuum - for instance, it's consistent that the constructible reals form such a set. We just can't prove that they do, so there's no problem. – Noah Schweber Nov 12 '17 at 20:42
  • @NoahSchweber ah ok... do you know to what degree their relationship to $\aleph_1$ is nailed down? It would seem logical to say (in layman's terms) that as far as ZFC can see, they're the same. But as far as ZFC can see they equal $\aleph_2$ as well. Is there some reasonable theory that can see more? – it's a hire car baby Nov 13 '17 at 12:30

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