Tim Tyler wrote:
> Anon.  wrote or quoted:
> 
>>Tim Tyler wrote:
>>
>>>Anon.  wrote or quoted:
>>
> 
>>>>>>>It's like claiming that half the integers are even.
>>>>>>
>>>>>>Err, they are.  There are just rather a lot of them.
>>>>>
>>>>>No, there aren't.
>>>>>
>>>>>There are an infinite number of even numbers.
>>>>>
>>>>>There are an infinite number of odd numbers.
>>>>>
>>>>>Divide infinity by infinity and the result is indeterminate.
>>>>
>>>>If there are an equal number of even and odd numbers, then
half of the 
>>>>numbers must be even.
>>>
>>>This is not true when the sizes of the sets involved are infinite.
>>
>>But they're the same size!  We can count them!
> 
> 
> That doesn't help - since the sizes are infinite - and 
> one infinite number divided by another one does not
> necessarily equal 0.5.
> 
> 
>>>>This must be true because for every even number, I can add
1 and get an 
>>>>odd number.  Conversely for every odd number I can add 1
and get an even 
>>>>number.  Hence, by the operation of adding 1, I can produce an even 
>>>>number for every odd number and vice versa.  Ergo, half of
all numbers 
>>>>are even, and half are odd.
>>>
>>>I can easily create a map between every even number an 5 unique odd 
>>>numbers - i.e I can map from 2x to 5x+1, 5x+3, 5x+5, 5x*7 and 5x+9.
>>
>>Yes, but that's not the operation of adding 1 is it?
> 
> 
> I never said it was.
> 
> What it proves that - as well as there being one odd number for every 
> even number there are also five odd numbers for every even number.
> 
> That's not good news for the assertion that the ratio of the number of 
> even numbers to the number of odd numbers is one.  Much the same argument 
> will "prove" the ratio is anything you care to mention.
> 
I think you're missing my point - that one can create a one to one 
correspondence between each even and odd number by the process of adding 1.

Others seem to think that I'm doing something wrong, and I may well be, 
in which case I'd like to know what the problem is - email to me (not 
the list)!



>>>Simply beacuse ratios of the sizes of infinite sets make little 
>>>mathematical sense, that does not render all notions of probability 
>>>useless.
>>
>>You are claiming this, but I have yet to see any proof. [...]
> 
> 
> Probably because this is sci.bio.evolution :-(
> 
> If you are *still* in doubt, look up:
> 
> "Classical definition of probability"
> 
> ...and...
> 
> "Frequency definition of probability"
> 
> ...or more simply, just take my word for it that probability can be quite 
> constently defined as a limit as the number of samples or trials tends to 
> infinity - and let the matter drop.
> 
As long as you admit that you have heard of Kolmonogorov's definition of 
probability, which does not depend on a limit argument.  To get back to 
my original point, this means that one can define proportions from 
infinite populations - Kolmonogorov defines the set that he constructs 
his measure so that it can be countably infinite.

Yes, probability (and proportion) _can_ be defined as a limit, but there 
are other ways of doing this, and so an insistence on the necessity of a 
limit argument is incorrect - that's the only point I was trying to make.

Bob

-- 
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
Fax:  +358-9-191 22 779
WWW:  http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org
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