How many integer sequences?
Michael Somos
somos at grail.cba.csuohio.edu
Sun Oct 24 05:35:15 CEST 1999
On Fri, 22 Oct 1999, Jud McCranie wrote:
> How many integer sequences are there in terms of Alephs? Aleph-1?
This sounds like a FAQ. In fact, it is important to precisely define what
a sequence is for purposes of the EIS. However, that is a long and detailed
essay which I am still working on. In this case, it is possible to give a
short answer using continued fractions. Each positive irrational real number
is uniquely associated with its continued fraction. The continued fraction
is determined by an infinite sequence of positive integers. We can encode
an arbitrary integer with a positive integer using the sequence [0,1,-1,2,
-2,3,-3,...]. Thus the infinite sequence of integers has the cardinality of
the continuum. There are other ways to encode infinite integer sequences as
a real number, and there are other ways to determine the cardinality of the
infinite sequences, but all we need is at least one. Shalom, Michael
--
Michael Somos <somos at grail.cba.csuohio.edu> Cleveland State University
http://grail.cba.csuohio.edu/~somos/ Cleveland, Ohio, USA 44115
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