On Fri, 15 Feb 2008 17:31:51 -0800, Mark Dickinson wrote:
> On Feb 15, 7:59 pm, Steven D'Aprano <[EMAIL PROTECTED]
> 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
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