Turn off ZeroDivisionError?

[Steven D'Aprano]

On Fri, 15 Feb 2008 17:31:51 -0800, Mark Dickinson wrote:

> On Feb 15, 7:59 pm, Steven D'Aprano <st... at REMOVE-THIS-
> cybersource.com.au> wrote:
>> On Fri, 15 Feb 2008 14:35:34 -0500, Steve Holden wrote:
>> >> I don't understand:  why would +INF not be equal to itself?  Having
>> >> INF == INF be True seems like something that makes sense both
>> >> mathematically and computationally.
>> >>  [...]
>>
>> > There are an uncountable number of infinities, all different.
>>
>> But the IEEE standard only supports one of them, aleph(0).
>>
>> Technically two: plus and minus aleph(0).
> 
> Not sure that alephs have anything to do with it.  And unless I'm
> missing something, minus aleph(0) is nonsense. (How do you define the
> negation of a cardinal?)

*shrug* How would you like to?

The natural numbers (0, 1, 2, 3, ...) are cardinal numbers too. 0 is the 
cardinality of the empty set {}; 1 is the cardinality of the set 
containing only the empty set {{}}; 2 is the cardinality of the set 
containing a set of cardinality 0 and a set of cardinality 1 {{}, {{}}}
... and so on.

Since we have generalized the natural numbers to the integers

... -3 -2 -1 0 1 2 3 ...

without worrying about what set has cardinality -1, I see no reason why 
we shouldn't generalize negation to the alephs. The question of what set, 
if any, has cardinality -aleph(0) is irrelevant. Since the traditional 
infinity of the real number line comes in a positive and negative 
version, and we identify positive ∞ as aleph(0) [see below for why], I 
don't believe there's any thing wrong with identifying -aleph(0) as -∞.

Another approach might be to treat the cardinals as ordinals. Subtraction 
isn't directly defined for ordinals, ordinals explicitly start counting 
at zero and only increase, never decrease. But one might argue that since 
all ordinals are surreal numbers, and subtraction *is* defined for 
surreals, we might identify aleph(0) as the ordinal omega ω then the 
negative of aleph(0) is just -ω, or {|{ ... -4, -3, -2, -1 }}. Or in 
English... -aleph(0) is the number more negative than every negative 
integer, which gratifyingly matches our intuition about negative infinity.

There's lots of hand-waving there. I expect a real mathematician could 
make it all vigorous. But a lot of it is really missing the point, which 
is that the IEEE standard isn't about ordinals, or cardinals, or surreal 
numbers, but about floating point numbers as a discrete approximation to 
the reals. In the reals, there are only two infinities that we care 
about, a positive and negative, and apart from the sign they are 
equivalent to aleph(0).


> From the fount of all wisdom: (http://en.wikipedia.org/wiki/
> Aleph_number)
> 
> """The aleph numbers differ from the infinity (∞) commonly found in
> algebra and calculus. Alephs measure the sizes of sets; infinity, on the
> other hand, is commonly defined as an extreme limit of the real number
> line (applied to a function or sequence that "diverges to infinity" or
> "increases without bound"), or an extreme point of the extended real
> number line. While some alephs are larger than others, ∞ is just ∞."""

That's a very informal definition of infinity. Taken literally, it's also 
nonsense, since the real number line has no limit, so talking about the 
limit of something with no limit is meaningless. So we have to take it 
loosely.

In fact, it isn't true that "∞ is just ∞" even in the two examples they 
discuss. There are TWO extended real number lines: the projectively 
extended real numbers, and the affinely extended real numbers. In the 
projective extension to the reals, there is only one ∞ and it is 
unsigned. In the affine extension, there are +∞ and -∞.

If you identify ∞ as "the number of natural numbers", that is, the number 
of numbers in the sequence 0, 1, 2, 3, 4, ... then that's precisely what 
aleph(0) is. If there's a limit to the real number line in any sense at 
all, it is the same limit as for the integers (since the integers go all 
the way to the end of the real number line).

(But note that there are more reals between 0 and ∞ than there are 
integers, even though both go to the same limit: the reals are more 
densely packed.)



-- 
Steven