NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/11046

Text File | 1992-09-07 | 2.8 KB | 55 lines

Path: sparky!uunet!olivea!mintaka.lcs.mit.edu!bloom-picayune.mit.edu!athena.mit.edu!zeno
From: [e-mail] (Richard Duffy)
Newsgroups: sci.math
Subject: equivalents of C.H. (was: Beloved Books + Request)
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Date: 6 Sep 92 00:52:48 GMT
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In article <[e-mail]>, Allan Adler writes:
>
> One of my favorite books is Sierpinski's Hypothese du continu, which studies
> the continuum hypothesis and its consequences and equivalents in analysis.

Ah, one of my favorites as well.  I believe it's even still in print from
Chelsea (N.Y.), though I forget whether it's the English translation that
they have.

Another of the results in that book is:  CH <==> there is a real function  f
such that the plane is a countable union of sets, each being either a
translate of the graph of  f  or a translate of  f's  graph rotated 90 degrees.
(Clearly we're talking about an extremely discontinuous function here).  I
would like to improve this result, if possible, to make the plane-covering
union be *disjoint*, but I have a notion this won't be possible ... any
guesses?

I can't refrain from bringing up again (though I haven't mentioned it here
in a long time) the following, which I happened across quite by chance once
when looking at another paper in some bound journal volume:

This is one of Paul Erdos's many delightful oddments: CH is equivalent to
the existence of an uncountable family  F  of entire functions (complex
functions holomorphic on the whole plane) such that for all points  z , the
set of images  {f(z) : f  in  F}  is countable.  Think about this in light of
the fact that any two distinct entire functions must take different values
at all but finitely many points in {1, 1/2, 1/3, 1/4, ...}, for example, yet
for each  1/n  the uncountably many  f's  take on only countably many values
f(1/n).  Pretty peculiar.     This equivalence is distinguished by the fact
that its proof (which is pretty short) uses bona-fide facts about convergent
power series---it's not just set theory in disguise.  It's not even
*real-variable* theory in disguise: there is an explicit uncountable family
of infinitely differentiable functions from  R  to  R  whose set of images at
each real number has cardinality at most 2! (an amusing exercise).



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