Putting the following three statements together:(adsbygoogle = window.adsbygoogle || []).push({});

(a) Assuming that the continuum hypothesis is false, the power of the continuum 2^{[itex]\aleph0[/itex]}is real-valued measurable.

(b) The existence of a real-valued measurable and the existence of a measurable (= real-valued measurable & inaccessible) cardinal are equiconsistent.

(c) If there exists a measurable cardinal, the continuum hypothesis is false.

it sounds like this would imply the following absurd statement:

(d) Assuming that the continuum hypothesis is false, the existence of 2^{[itex]\aleph0[/itex]}and the existence of a measurable cardinal are equiconsistent.

What is wrong? Is (a) incorrect, or am I putting these together wrong? If (a) is incorrect, is there a clear example to show why?

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# Questions about real-valued measurable cardinals and the continuum

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