Re: infinity & set membership

[nobody@huge.cajones.com (Huge Cajones Remailer)]

At 4:36 PM 1/19/1997, Jim Choate wrote:
>"The least upper bound of a set of real numbers is often called the
>supremum, the greatest lower bound its infimum. In general the supremum and
>infimum of a set ARE MEMBERS OF THE SET or at least LIMITS OF SEQUENCES OF
>MEMBERS OF THE SET."
>
>[capitalization is mine]
>
>In this case the suprenum is infinity.
>
>Introduction to Calculus and Analysis
>Courant and John
>Vol. 1, pp. 97, Section e.
>1965 Edition
>Library of Congress: 65-16403

I presume this is a reference to Jim's earlier post:

At 1:19 PM 1/19/1997, Jim Choate wrote:
>Forwarded message:
>
>> Date: Sun, 19 Jan 1997 12:20:45 -0600
>> From: Sir Robin of Locksley <tozser@stolaf.edu>
>
>>  >Is it possible to prove that number 0.1234567891011121314151617181920...
>>  >iz irrational? 
>>  
>> Most definately. All you need to do is prove that the set of this number is
>> uncountable, ergo is irrational. If you have friend who have done math in real
>> analysis they can explain more.
>
>irrational means 'not expressible as the ratio of two integers', this does
>not imply uncountable.
>
>Whether 0.12345678910111213... is irrational or not depends on how we choose
>to define a rational number where the denominator and numerator are both
>approaching infinity and how quickly those approaches occur.
>
>[ What is your definition of infinity/infinity ?]
>
>As alluded to before,
>
>  0.12345678910111213    =    12345678910111213.... / n 
>
>  (where n = infinity)
>
>0.123456789101112... is certainly countable because by the definition of
>countable it must be 1-to-1 with the counting numbers (ie non-negative
>integers), which this one clearly is since it contains each positive
>non-zero integer (ie the set of numbers required to produce the number are
>clearly less, 1 less to be exact, than the counting numbers).

Of course this same argument could be used to claim that pi is
also rational.  pi =  314159.... / n  (where n=infinity).

What could be the problem here?  One problem might be that an integer
(not including infinity) divided by "infinity" should be 0.

The p/q definition of rational numbers does not rely on "infinity"
being a member of the set of integers.  Also, we rarely hear this
dialogue: "Give me an integer."  "Okay - infinity!"

Still, let's define a set which contains the integers as well as
positive and negative "infinity".  I will call this set the
"Choate-integers."

Let's add some reasonable rules for working with +inf and -inf.  n is
a non-zero positive traditional integer.

a. -inf = +inf * -1
b. +inf * n = +inf
c. +inf + n = +inf
d. +inf / +inf = 1

Operations involving multiplication or division of +inf and zero are
undefined.

Using these definitions, you will find that the p/q definition of
rational numbers operates as expected on the set of Choate-integers.

The argument I supplied earlier for the rational nature of 0.1234...
still holds.

Math Man