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MathGroup Archive 2006

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Re: Re: )


On 9 May 2006, at 15:35, Maxim wrote:

> On Sat, 6 May 2006 06:20:52 +0000 (UTC), Andrzej Kozlowski
> <akoz at mimuw.edu.pl> wrote:
>
>>
>> laws of arithmetic do not hold). Some of them are hard to explain: I
>> can't see any good reason at all why Infinity^Infinity is
>> ComplexInfinity, and it seems to contradict the the most basic rule
>> that x^y is always real when x and y are positive reals. Besides, as
>> I mentioned earlier, Infinity and ComplexInfinity do not belong
>> together in any topological model known to me (you need a
>> "topological model" to be able to consider the issue of continuity)
>> and should never appear in the same formula. I can only consider this
>> as a bug, and a rather silly one.
>>
>
> I think this is as it should be: we need to consider all the sequences
> converging to Infinity, and, for example, Limit[(x + I*Pi)^x, x ->
> Infinity] == -Infinity. So in that sense Infinity^Infinity ==
> ComplexInfinity: when z = z1^z2 and z1, z2 go to Infinity the absolute
> value of z always tends to Infinity but the argument can be arbitrary.
>
> Maxim Rytin
> m.r at inbox.ru
>


One can perhaps make some sense of this even topologically if one  
assumes  that map {z,w}->z^w has as its target space the Riemann  
sphere and as its domain the Cartesian product of 2-discs. In other  
words it is not a map of the form  X x X -> X, as one usually  
supposes.  But then it is quite a different map from the one that gives

2^Infinity

Infinity

and so on. I certainly cannot conceive of any sensible topology on  
the set that is the union of the complex plane, the set of  
DirectedInfinities and the point ComplexInfinity, can you? Any such  
topology would have pretty  shocking properties.... And if there is  
no topology involved than what is Limit supposed to mean?  Still, I  
admit, that even if things that don't make any mathematical sense may  
sometimes be acceptable  in a symbolic algebra program, for purely  
practical reasons, but then they are pretty likely to lead to  
confusion and contradictions. This is in fact the current state of  
affairs in this regard and it makes me feel that the whole  
Mathematica approach to this business of Infinities  maybe misguided.

Of course,  as we are not talking about mathematics or any kind of  
empirical science this is all ultimately a matter of taste and as we  
well know there is no accounting for tastes... or sense of humour.

Andrzej Kozlowski

  


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