Anon.  wrote or quoted:
> Tim Tyler wrote:
> > Bob O'Hara  wrote or quoted:
> >>Tim Tyler wrote:

> >>>>>Popularisers should make explicit the behaviour is
what happens as
> >>>>>the population size tends towards infinity - and
not attempt to pass
> >>>>>it off as an effect in an infinite population.
> >>>>
> >>>>But it is - in finite populations, you get an excess
of homozygotes, as 
> >>>>any student of population genetics should know.
> >>>
> >>>Any mention of gene frequencies in an infinite population
is nonsense -
> >>>as I stated originally.
> >>>
> >>>You can't talk about a fraction of an infinite population having
> >>>a trait.  You would get different results for that
fraction depending
> >>>on how you enumerated through the population.
> >>
> >>I don't understand what you mean, but by that argument, you can't even 
> >>define a fraction or a probability.
> >
> > Fractions have nothing to do with infinite sets.
> 
> But there are an infinite number of fractions, so they have at least 
> that to do with infinite sets.

You can talk about fractions just fine without ever mentioning infinite
sets.

It is incorrect to say that you can't define fractions without
reference to infinite sets.

A fraction is just one finite number divided by another one.

> >>>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.

The ratio of two infinities is either another infinity, an infinitessimal,
or is undefined - depending on the infinities in question.

> 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.

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.

You simply can't argue like that about ratios between the sizes of
infinite sets.

> >>>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.

If you ask what proprtion of rational numbers is smaller than 1/3
the answer is *not* "one third of them".  The answer is that the
question makes no sense - because it is a ratio of two infinite
numbers, and - in mathematics - oo / oo is not defined.

> >>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.

Simply beacuse ratios of the sizes of infinite sets make little 
mathematical sense, that does not render all notions of probability 
useless.

> >>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.

> >>Infinity is a difficult concept (I know - there are lots of it I don't 
> >>understand), so I think one should be cautious about making any 
> >>pronouncements on it unless one is sure about what mathematics
> >>does and does not say on the subject.
> > 
> > How is that relevant?
> 
> You're trying to argue about the use of infinity.  I'm pointing out that 
> one should be careful when doing this.  This seems relevant.

OK.

> > 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

This result is codified in things such as the IEEE 754 standard.
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