NetNews Usenet Archive 1992 #18/NN_1992_18.iso/spool/sci/math/10307

Text File | 1992-08-16 | 2.6 KB | 49 lines

Newsgroups: sci.math
Path: sparky!uunet!cis.ohio-state.edu!magnus.acs.ohio-state.edu!wjcastre
From: [e-mail] (W.Jose Castrellon G.)
Subject: Re: A Non-Cantorian Set Theory question
Message-ID: <[e-mail]>
Sender: [e-mail]
Nntp-Posting-Host: bottom.magnus.acs.ohio-state.edu
Organization: The Ohio State University,Math.Dept.(studnt)
References: <[e-mail]>
Date: Sun, 16 Aug 1992 21:36:57 GMT
Lines: 36

In article <[e-mail]> [e-mail] writes:
>Quite a while ago, I read Martin Gardner write in one of his _Mathematical
>Games_ column in _Scientific American_ that on a plane, a letter like _O_ can
>be written--allowing smaller O's to be written inside larger O's-- _c_
>times where _c_ is the cardinality of the continuum. On the other hand, he
>pointed out, letters like _T_ can only be written aleph-nought times.
>
>A question that has remained in my mind for a long time since is the following
>
>Since we know that non-Cantorian set theories are possible, is there a
>one-dimensional shape which can be written some aleph times where that aleph
>is between aleph-nought and _c_? If so, what would the shape be?
>
>I have once asked my mathematics teacher about it, and he remarked that that
>shape might be a fractal!
>
>Does anyone have any idea on this?

Nice question. If you allow disconnected spaces, the answer is yes:
Let A be a subset of reals of cardinality  K  and call G the set obtained by
taking all finite sums of elements in A and their negatives, then G also has
cardinality K. Now partition the reals into equivalence classes, such that
two elements belong to the same equivalence class iff their difference is an
element of G. If we form a set S by picking one element from each equivalence
class, then the real line is the disjoint union of K-many sets, each congruent
to S  (Just take S_u = {s+u : s belongs to S}, for each u in G). Now by using
Zorn's lemma, S can be constructed such that no more than K-many sets similar
(i.e. homotetic) to  S  can be disjoint in the real line. To get the final sub-
set of the plane, it is enough to give  S some thickness, i.e. make P equal to
the cartesian product of  S  and the interval [0,1], for example (though
lengthy to include in here, all details are easy to handle, just as for the T).
I believe it is possible to construct a connected set P too, but I havent found
a proof of that yet.

As for well-behaved spaces, I don't see any unreasonable obstacle to prove
that for P path-connected the answer is either one of Aleph_0 or 2^Aleph_0.