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On 10/11/2010 9:55 AM, Darren New wrote:
>>> Funny enough, infinities make things much more complicated. That's why
>>> virtually every bit of math deals with infinities.
>>
>> Really? I had no idea that was true.
>
> Sure. Almost all functions of interest have domains of infinite size,
> for example. Otherwise, you'd just list out what the function returns
> for each possible input and there wouldn't be any interesting abstract
> questions left about it.
Two comments:
1) There'd still be interesting questions to ask about partial
functions, since I'm pretty sure the restriction of Rice's theorem to
functions between finite sets still works (when the range and domain
both have at least two elements, obviously). It only gets
`uninteresting' when you restrict yourself to a finite set of
*programs*, not functions.
2) I don't agree with your view on the use of infinity in mathematics.
On the contrary, infinity (or rather unboundedness, which I assume is
that you meant) is often used as a way to *simplify* things. I think
that "big O" notation is an excellent example of this, but there's
plenty of others. Rather than type a bunch about it here, I managed to
find a nice blog post about it:
http://www.johndcook.com/blog/2010/09/09/infinite-is-easier-than-big/
>>>> So it seems that the fallacy of solving the Halting Problem is in
>>>> Magic() being able to predict the outcome of an infinite number of
>>>> cycles of work with some finite number of cycles of processing.
>>>> Interesting...
>>>
>>> Yes, exactly.
>>
>> When you state it like that, it becomes "well *duh*, of *course* you
>> can't do that!"
>
Except when you can.
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