FOM: Challenge on CH
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Mon Dec 22 18:10:44 EST 1997
I have argued both for the meaningfulness and definiteness of CH, and also for
its "unreality" in the sense that our intuitions and theorems about it do not
correspond to anything in the physical world. But I don't think I'm thereby
forced into the corner of admitting that CH is undecidable in any essential
sense. One point I would like to remind everyone of is that we DO know
something about the cardinality of the continuum. Namely, we know that
1) it is uncountable
2) it is not the union of countably many smaller cardinals!
In fact 2) has essentially the same (easy) proof as 1) except that you need a
weak version of the axiom of choice to select for each of the countably many
injections from the smaller cardinals into c an element not in the range.
In what way is our knowledge that c is not the union of countably many smaller
cardinals meaningless, indefinite, or unreal? Who will deny that this is
something we actually *KNOW* about the size of the set of reals? Claims that we
can't ever know which aleph c is need to explain why we CAN know THIS.-J Shipman
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