Kevin Wampler wrote:
> inputs I'm actually having trouble thinking of a case where this is true 

I don't know what "this" you're talking about. Your comments don't make 
sense in the context of the bit you quoted.

> My example with big-O notation was actually slightly different than 
> that.  

Right.

> Thus saying that every program on a finite 
> machine finishes in O(1) times is sort of disallowing infinity in one 
> area while letting it slip by in another.

Basically, yes.

> My point with big-O notation was instead that the allowance of 
> arbitrarily large inputs in the definition is actually massively 
> simplifies the problem.  

I didn't mean to say it didn't. I thought I had only said it was more 
interesting with infinities.

> I'm not entirely sure that we actually have different viewpoints on 
> this, and it's possible that I'm just misunderstanding your argument. 
> I'm mostly basing my assumptions about your position on your comment:
> 
>> Funny enough, infinities make things much more complicated. That's why
>> virtually every bit of math deals with infinities.
> 
> Which seems to run counter to my experience, 

In many cases where everything is finite, the math becomes trivial, because 
math doesn't care how big something is if it's bounded.

I probably should have added an "often" in there. Infinities often make 
things more complicated.

> where I'm used to seeing 
> unbounded values allowed as a simplifying assumption, and see them as 
> common because it's such a powerful tool for allowing general statements 
> about whatever branch of math you're considering.

Sure. I didn't mean using infinities in the analysis always makes things 
more complicated. I meant using infinities in the problem statement makes 
things interestingly complicated, as opposed to problems where everything is 
finite, everything halts, etc.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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