You have an ink stamp that is so amazingly precise that, when inked and pressed down on the plane, it makes every circle whose radius is an irrational number (centered at the center of the stamp) black.

Is it possible to use the stamp three times and make every point in the plane black?

If it is possible, where would you center the three stamps?

Am I mistaken, or was it sporadically bandied about in the below discussion that the cardinality of the irrationals is Aleph-1?

Last I heard, no one knew where the cardinality of the irrationals fell on the aleph scale. All that was known was that the irrationals were equivalent to the power set of the rationals (2^Aleph-0).

In fact, I thought that the Continuum Hypothesis, (the assertion that 2^Aleph-0 = Aleph 1) was known to be undecidable from the axioms of ZFC.

Posted by Avin on 2005-05-18 14:11:51