# Cardinal characteristics

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)?

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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 '14 at 14:23

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.