I was wondering whether it is consistent to have $\frak{c} = \aleph\_{\frak{c}}$ where $\frak{c} = 2^{\aleph\_0}$ is the cardinality of the reals (over ZFC). If so, what interesting consequences of this statement are known (besides ¬CH)? I was curious about this because in some sense $\frak{c}$ is the largest possible number of cardinals below $\frak{c}$, and this is partly motivated by the idea that $\frak{c}$ may be so large as to be 'unreachable' via approximation by fewer smaller cardinals, which seems similar in nature to [an opinion of Cohen on CH](https://mathoverflow.net/a/265083).


From what I have read, I think that it is consistent (relative to ZFC) for $\frak{c}$ to be the $ω\_1$-th fixed-point of $\aleph$, which would be one possibility satisfying $\frak{c} = \aleph\_{\frak{c}}$. But can $\frak{c}$ be the $\frak{c}$-th fixed-point of $\aleph$, and does this yield even more interesting consequences?



What if $\mathbb{R}$ is in bijection with the cardinals less than $\frak{c}$?
Yes. Start with a model of $\sf CH$, then take the least fixed point with uncountable cofinality. Call that $\kappa$. Now add $\kappa$ Cohen reals.


Since fixed points form a club of ordinals, you can iterate the fixed points enumeration. Repeat that $\omega\_1$ times, then take the least one of cofinality $\omega\_1$ in that club. Now call that $\kappa$, and add that many Cohen reals.




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Assuming no large cardinals get involved, that means that $\frak c$ is singular. This by itself implies that Martin's Axiom fails, and that Cichon's diagram is not trivial, since some of the cardinal characteristics are provably regular.


Other than that, I don't believe we can say a lot more without adding more assumptions on the universe.


