# Re: mathematical infinity (Was Bypassing s322)

```'Infinity' turns out not to have a single definition. (Learned this in high
school (!) for a math competition.) The 'smallest' infinity, named
Aleph-sub-zero or Aleph-null by Georg Cantor, represents the set of all
integers. There are no orders of magnitude for Aleph-null. Any set of integers
starting at an arbitrary point is represented by Aleph-null. For example, the
set of all even integers, the set of all odd integers, the set of all negative
and positive integers together, the set of all integers divisible by 3 or 3000
or 3000000. All represented by Aleph-null. ```
```
There is at least one other 'infinity' that is larger than Aleph-null: the set
of all real numbers between any two arbitrary points, dubbed Aleph-sub-one. So
the number of real numbers between zero and one or between 1 and 1000000 are
both represented as Aleph-sub-one.

How this relates to z systems I don't know. Just showing off what I remember
from a hundred years ago. ;-)

.
.
J.O.Skip Robinson
Southern California Edison Company
Electric Dragon Team Paddler
SHARE MVS Program Co-Manager
323-715-0595 Mobile
626-302-7535 Office
robin...@sce.com

-----Original Message-----
From: IBM Mainframe Discussion List [mailto:IBM-MAIN@LISTSERV.UA.EDU] On Behalf
Of Walt Farrell
Sent: Saturday, September 17, 2016 5:25 AM
To: IBM-MAIN@LISTSERV.UA.EDU
Subject: (External):Re: mathematical infinity (Was Bypassing s322)

On Fri, 16 Sep 2016 21:20:59 -0500, Joel C. Ewing <jcew...@acm.org> wrote:

>Obviously the description of floating-point multiply given above needs
>some liberal interpretation as well.  The hardware obviously cannot
>produce an "infinite product". Since there is a fixed maximum magnitude
>positive or negative that can be represented in each of the various
>number representations used on any computer, the maximum product that
>could be attempted would be the square of that magnitude.  As a
>practical matter, any result that overflows that maximum magnitude may
>be regarded as infinity by the architecture, or it might just produce a
>garbaged value and an overflow indication.  Some architectures have
>reserved values that stand for +∞  and -∞ so that once one of these
>values is produced in a chain of computations, subsequent calculations
>based on those values also convey that the results have lost meaning.

From the PoP it seems that our hardware can deal with the concept of infinity
(I think that's part of the IEEE floating-point standard) :
<quote>
Infinities

BFP and DFP data include an infinite numeric datum, called infinity. Infinities
can participate in most arithmetic operations and give a consistent result,
usually infinity. An infinity has a sign bit. In comparisons, infinities of the
same sign compare equal, +∞ compares greater than any finite number, and -∞
compares less than any finite number.
</quote>

--
Walt

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