FOM: Re: Challenge on CH

Harvey Friedman friedman at
Mon Dec 22 15:07:37 EST 1997

Reply to Shipman 6:10PM 12/22/97.

>  In what way is our knowledge that c is not the union of countably many
>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

I call attention to my 12:54PM 12/17/97, Re:Reals and reality, and my
2:28AM 12/8/97 on the continuum hypothesis. With a complete picture of,
say, V(omega + omega), one easily derives such "facts." But we also may be
able to prove that no complete picture of, say, V(omega + omega) suffices
to answer the continuum hypothesis. And we may come to accept the idea that
in this kind of context, we only know things from complete pictures, and
not by any other method. If so, then results that demonstrate the inability
of any complete picture to solve the continuum hypothesis would become
definitive, and solve your dilemma. In fact, the reason I find this
appraoch so attractive is that it "solves" such dilemmas, which otherwise
cause serious difficulties.

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