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Assuming Continuum hypothesis is not true, How many cardinals $k$ exist which are $\aleph_1 < k < \mathfrak c$? Can I assume that there is a finite number of these cardinals or is there an infinite number (like ordinals). Is there a common topology on these cardinals (like, for example, the order topology on ordinals)? enter image description here

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I don't see why this is about topology. If this is simply a question "For what values of $\alpha$, it is consistent to have $2^{\aleph_0}=\aleph_\alpha$?", then we have an answer for that already. If you are trying to understand the diagram, note that hardly any inequality in the diagram is strict. So you might be asking how many of these can have different values, or you might be asking if it is consistent that each node in the diagram you added has a different value. But the text of the question still asks the very simple question about the possible values of $\frak c$. –  Asaf Karagila Sep 1 at 14:23

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The number of cardinals between $\aleph_1$ and $\mathfrak c$ can be anything. In particular, it can (consistently with ZFC) be any finite number you like, or it can be countably infinite, or it can be uncountably infinite. It can even be $\mathfrak c$ itself. These facts come from a paper of Solovay, "$2^{\aleph_0}$ can be anything it ought to be," written very soon after Cohen proved the independence of CH.

The title of your question and the diagram you included suggest that you might not be so interested in arbitrary cardinals but rather in those that can reasonably be defined as cardinal characteristics of the continuum. Even in this case, the number of such cardinals can be infinite (or can be a large finite number). A relevant reference here is the paper "Many simple cardinal characteristics" by Golldstern and Shelah.

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I see. Thank you! –  Shir Sivroni Sep 1 at 14:34
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You probably wanted to write "Many simple cardinal invariants" and not "Many simple cardinal characteristics". DOI: 10.1007/BF01375552, arxiv, Google. –  Martin Sleziak Sep 1 at 15:13
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@MartinSleziak Yes; I think of the two as synonymous, but that doesn't count in titles. –  Andreas Blass Sep 1 at 15:16
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To complement Andreas answer, on the other hand, if you are interested on the cardinals in Cichon's diagram, it is still open whether we can separate them all. There is ongoing work on this (for instance, look at recent papers by Vera Fischer and her collaborators), that requires refining the common iteration techniques used in forcing. –  Andres Caicedo Sep 1 at 15:51
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@Andres: And by Vera's husband, Arthur Fischer and his coauthors, where they separate five cardinal characteristics at once. Needless to say, the set theory world is waiting for some Fischer-Fischer papers. –  Asaf Karagila Sep 4 at 8:25

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