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It is written on Wikipedia:

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

Suppose that CH is not true, that is that there is such a set, call it $C_1$, so, $|\mathbb Z|<|C_1|<|\mathbb R|$

But, does some principle "forbids" us to suppose that there are also sets $C_2$ and $C_3$ so that we have $|\mathbb Z|<|C_2|<|C_1|<|C_3|<|\mathbb R|$, and $4$ more sets with cardinalities between those $5$, and so on and so on...

That is, if CH is not true, can there be an infinitely countable number of sets $S_i$ between $\mathbb Z$ and $\mathbb R$ so that we have $|\mathbb Z|<| S_1|<...<|S_{\infty}|<|\mathbb R|$. And, if there cannot be an infinitely countable number of sets between $\mathbb Z$ and $\mathbb R$, all with different cardinalities, can there be a finite number of them (a finite number greater than $1$)?

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Grešnik
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  • Yes, yes, and yes. The number of different cardinalities between $|\mathbb Z|$ and $|\mathbb R|$ could be $0$, or $1$, or $42$, or $|\mathbb Z|$, or even $|\mathbb R|$. It can't be more than $|\mathbb R|$. – bof Jun 21 '19 at 09:51
  • @bof Why it can´t be more than $|\mathbb R|$?, That ain´t obvious to me. Although, I bet that you have a simple argument why it cannot be. – Grešnik Jun 21 '19 at 09:54
  • Because then there are more than continuum cardinalities below the cardinality of the continuum, and the cardinality of the continuum contains all those smaller cardinals. – Vsotvep Jun 21 '19 at 10:06
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    Perhaps more interesting is that there could be $\aleph_\kappa$ many cardinalities below the continuum for arbitrarily large ordinal $\kappa$, due to Easton's theorem – Vsotvep Jun 21 '19 at 10:08
  • @Vsotvep But how it can be that it is allowable within ZFC that the number of different cardinalities between $\mathbb Z$ and $\mathbb R$ could be $|\mathbb R|$?. For example, if I prove that that number is not $|\mathbb R|$, would that be a new result in the field? – Grešnik Jun 21 '19 at 10:10
  • @Grešnik It is indeed known to be consistent that there can be $\vert\mathbb{R}\vert$-many cardinals between $\vert\mathbb{Z}\vert$ and $\vert\mathbb{R}\vert$ (that is, it's not just that no proof of impossibility is known, but we can prove that no such proof exists - unless ZF itself is inconsistent). Specifically, it is consistent that $\vert\mathbb{R}\vert$ is the $\omega_1$th fixed point of the $\aleph$-function. So you cannot prove that that number is not $\vert\mathbb{R}\vert$, unless you prove that all of ZF is inconsistent already. – Noah Schweber Jun 21 '19 at 18:46
  • @NoahSchweber Thank you Noah for the information, but, I think that I will try to find a proof from time to time. However, it may be that I will have to add at least one axiom to ZFC in order to prove that that is not possible. This sounds like cheating, but, if I can find some reasonable axiom, why not? – Grešnik Jun 21 '19 at 18:51
  • Well, I do not see any big problem if ZF is inconsistent. Or, at least, ZFC. – Grešnik Jun 21 '19 at 18:52
  • @Grešnik If ZFC is inconsistent then so is ZF - this was proved by Godel. Put another way, AC is a "safe addition" to ZF - just like CH (and AC+CH, and AC+GCH, and ...). ZF, ZFC, ZFC+CH, ZFC + "$2^{\aleph_0}$ is the $\omega_1$th $\aleph$-fixed point," and a bunch of other variations are all equi-consistent. – Noah Schweber Jun 21 '19 at 18:59
  • Obviously you're welcome to try to prove whatever you want, but if you're really interested you should study the consistency results and techniques (namely, constructibility and forcing) that have been mentioned here. They're quite hard, but understanding them (specifically, forcing) will clarify why the statement you're interested in can't be proved in ZFC alone. – Noah Schweber Jun 21 '19 at 19:01
  • @NoahSchweber I know that it is known that Godel had major impact on set theory and mathematical logic, but, I did not read his works. Are you sure that there is no gap in his proofs? I can try to master advanced set theory, at least I can try if nothing else. – Grešnik Jun 21 '19 at 19:03
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    @Grešnik "Are you sure that there is no gap in his proofs?" Um, yes, extremely so: all these arguments are (to put it mildly) quite well-understood, there's nothing mysterious about any of them. All of the material described here is in Kunen's book, Jech's book, and various other sources. For what it's worth, in my opinion constructibility (what Godel did) is much easier to learn than forcing (what Cohen did), but forcing is far more versatile. – Noah Schweber Jun 21 '19 at 19:12
  • @NoahSchweber I asked one more set-theoretic question about an hour ago, if you can help you´re welcome. Thank you for friendly advices and suggestions. – Grešnik Jun 22 '19 at 06:37

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Easton's theorem in particular implies (although the result itself is probably easier to prove than the full theorem) that if you fix a model $V$ of ZFC+CH then for all cardinal $\kappa$ of cofinality $>\omega$, there is a forcing extension of $V$ that preserves cardinals and cofinalities where $2^{\aleph_0} = \kappa$ ( note that $|\mathbb{R}|=2^{\aleph_0}$).

Now if $\kappa = \aleph_\alpha$, the "number of cardinalities between $\mathbb N$ and $\mathbb R$" is $\alpha$ (or "$\alpha-1$" arguably, but for limit $\alpha$ one can safely say $\alpha$). In particular, for any ordinal of cofinality $>\omega$, this number can be $\alpha$.

Your last question in the comments is "can it be $|\mathbb R|$ ?", i.e. can we have $\alpha = \kappa$, that is, $\aleph_\kappa= \kappa$. This question is : is there a cardinal of cofinality $>\omega$ that is an $\aleph$-fixed point ? The answer is yes, as for any regular cardinal $\kappa$ there are (many) $\aleph$-fixed points of cofinality $\kappa$, see for instance here for a proof. So in conclusion : there can be $|\mathbb R|$-many cardinals between $\mathbb N$ and $\mathbb R$ (of course not more)

Maxime Ramzi
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    Easton? Cohen proved that. In his first paper on forcing! Solovay is also credited for what is probably the statement you are referring to. – Asaf Karagila Jun 21 '19 at 11:36
  • @AsafKaragila : you're probably right, but I don't know enough about the specific statements, and I knew about Easton's theorem, that's why I put it there – Maxime Ramzi Jun 21 '19 at 11:47
  • To elaborate on Asaf's comment if you assume $\mathsf{GCH}$ in the ground model $M$, then forcing with the poset $\mathrm{Fn}(\kappa\times\omega,2,\omega)$ of finite partial functions $\kappa\times\omega\to 2$ for some $\kappa$ of uncountable cofinality forces $|\Bbb R|=\kappa$ in $M[G]$ (note that this poset is ccc so forcing does not collapse cardinals) – Alessandro Codenotti Jun 21 '19 at 16:11