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.

Post a reply to this message