NetNews Usenet Archive 1992 #18/NN_1992_18.iso/spool/sci/math/10310

Text File | 1992-08-16 | 2.8 KB | 54 lines

Newsgroups: sci.math
Path: sparky!uunet!zaphod.mps.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]> <[e-mail]>
Date: Mon, 17 Aug 1992 01:58:04 GMT
Lines: 41

In article <[e-mail]> I write:

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 followin
>>
>>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.

I've just found the connected set  P  and the proof involves no ideas different
from the above. (IMO it makes a nice exercise in plane analysis).
WJCG