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!

>>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?

What I missed out was a uniqueness statement (i.e. that I can create all 
odd numbers uniquely from the even numbers).

> That is exactly the same sort of argument as the one you gave - yet it 
> indicates that there are *five* odd numbers for every even number.
> 
> What conclusion should one draw from this?
> 
> The correct conclusion is that the argument you gave is useless.
> 
Not quite, because I can resurrect it.  For each integer (for which 
there are a countably finite number), I can create a single unique even 
number (i.e. I multiply the integer by 2).  I can also create a single 
unique odd number: multiply by 2 and add 1.  There are therefore as many 
even numbers as integers, and as many odd numbers as integers, so half 
of the integers must be even.

>>>>>No serious mathematician can talk about fractions of
infinite sets and 
>>>>>expect to be taken seriously.
>>>>
>>>>But they do.
>>>
>>>No - not unless the fractions are "zero" or "one".
>>
>>Rubbish, unless you're denying the existence of fractions.  Fractions 
>>are fractions of an infinite set, because there is an infinite number of 
>>numbers between 0 and 1 (proof: take the reciprocal of every positive 
>>integer).
> 
> 
> 1/3 is *not* the ratio of the size of the set of numbers smaller than 1/3 
> and the size of the set of numbers greater than 1/3.  That ratio is 
> the ratio of two infinite numbers - and thus is not well defined.
> 
So you claim, but have yet to provide a proof.  This is mathematics, so 
you should be able to provide a proof of your assertion that one can 
never determine the value of the ratio infinity/infinity.

> 
>>>>It's how probability is defined as a concept.
>>>
>>>Probability is defined as a mathematical limit, as N approaches infinity.
>>>
>>>That uses a limit as a finite set increases in size - not a
fraction of an 
>>>infinite set.
>>>
>>>E.g. see:
>>>
>>>http://www.wordiq.com/definition/Probability
>>
>>This doesn't show that probability is defined as a limit - the nearest 
>>you get is in the section "Probability in mathematics",
where they use 
>>"one approach" to give an interpretation - essentially,
the frequentist 
>>approach.  Note that when they discuss Kolmonogorov's definition of 
>>probability as a measure, they make no mention of any limits.
> 
> 
> You *can* define probabilities in terms of limits - without reference to 
> infinite sets.
> 
You can, but you don't have to.  Kolmonogorov didn't, and that's the 
formal definition used nowadays in probability theory.  The wordiq.com 
website includes an article on Kolmonogorov's formalisation of 
probability theory.

> 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.  Kolmonogorov's 
formulation of probability works fine for (countably) infinite sets of 
events, and hence there have to a be (countably) infinite number of 
probablities in the measure (i.e. the probability that 1, 2, 3,... 
events occur).

> 
>>>>I have a colleague who even wrote mathematical papers about
fractions 
>>>>of uncountable sets.
>>>
>>>If you can show me, I should be able to tell you if they contain the 
>>>fallacy under discussion.
>>>
>>>Probably he doesn't do that at all - and instead uses a limit.
>>
>>This was (I think - my copy is at home) the paper:
>>
>>E. Arjas & E. Nummelin & R.L. Tweedie: Semi-Markov processes on a 
>>general state space -theory and quasi-stationarity. J. Aust. Math. Soc. 
>>(Series A) 30 (1980): 187 - 200.
> 
> 
> Apparently too inaccessible for me to examine on the basis of a esoteric 
> point in a usenet debate - unless you know where it is publicly accessible.
> 
Alas not.

>>>Are you suggesting I don't know what I am talking about?
>>>
>>>That is not the case.
>>
>>Your evidence for this is?
> 
> 
> My mathematical credentials may not be publicly accessible for your
> inspection - but I do have a degree in mathematics.
> 
> Here is the (basically correct) answer given on mathforum regarding
> ratios of infinite quantities.
> 
>   http://mathforum.org/library/drmath/view/53337.html
> 
And he agrees with me:

"There are sort of some different "sizes" of
infinities, so this means that a quotient that looks like infinity
over infinity can sometimes be  a real number, and sometimes it is
just infinity."

He claims that infinity/infinity _can_ be a real number.

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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