- #1
nomadreid
Gold Member
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Putting the following three statements together:
(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?
(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?