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I feel like I have a very simple proof that there can't exist any real measurable cardinals. I know the proof is wrong somewhere but I cannot find the mistake.

By a result from Ulam all real measureable cardinals must be weakly inaccessible. This then means that the cardinal sucessor of any measurable carindal is not real measurable (it is not a limit cardinal). But for $\alpha$ not a real measuable cardinal, all $\beta<\alpha$ must also not be real measurable (assume assume there is a measure $\mu$ on $2^\beta$ that satsisfies the real measurable property, then the measure $\nu(A) = \mu(A\cap \beta)$ on $2^\alpha$ would also satisfies the real measurable property [this is essential given in theorem 2, page 317 of Kuratowski and Mostowski's set theory book]). Does this not imply that if $\alpha$ is real measurable, then $\alpha+1$ is also not real measurable and so $\alpha$ cannot be real measurable?

daRoyalCacti
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    This is a comment, not an answer, because I'm far from expert on the subject. But I think your problem is mixing up two different definitions of "real-valued measurable cardinal". Define RVMC one way, and any cardinal greater than a RVMC (including its successor cardinal) is also a RVMC. Define RVMC the other (standard) way, and a RVMC must be weakly inaccessible, so its successor cannot be a RVMC. What is your definition of RVMC? – user14111 Apr 23 '24 at 04:35
  • @user14111 I'm saying that a cardinal is real valued measurable if there is a non-zero measure defined on the power set of the cardinal that is 0 on singletons. – daRoyalCacti Apr 23 '24 at 04:40
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    Do you require your measure to be $\kappa$-additive? If you do, then your construction doesn't work; if you don't, then Ulam's result doesn't apply. It may be easier to compare two similar notions of measurability: "there is a countably closed nonprincipal ultrafilter on $\mathcal{P}(\kappa)$" vs. "there is a $\kappa$-closed nonprincipal ultrafilter on $\mathcal{P}(\kappa)$." These are, perhaps confusingly, called "Ulam-measurability" and "measurability" respectively; measurable cardinals must be inaccessible but Ulam-measurables need not be and indeed the Ulam-measurables are closed upwards. – Noah Schweber Apr 23 '24 at 04:59
  • @NoahSchweber Thank you very much! I realize that I am mixing up definitions. – daRoyalCacti Apr 23 '24 at 05:17

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