Re: Date for Last Common Ancestor?

Stephen Barnard (
Fri, 16 Aug 1996 19:26:54 -0800

> In article <>, Stephen Barnard
> <> writes:
> >
> >CHESSONP wrote:
> >>
> >>
> >> An example. Let I_k be the interval of real numbers greater than -1/k
> and
> >> less than 1+1/k, or in more familiar notation, (-1/k, 1+1/k). This
> >> interval has length 1+2/k. If we consider the sequence of interval
> lengths
> >> {1+2/k} obtained from the associated sequence of intervals {I_k}, we
> see
> >> that the lengths are strictly decreasing. However, the sequence of
> lengths
> >> does not go to zero, but rather to a non-zero number, namely one.
> >
> >I don't see how this applies to my argument. Your example operates in an
> >entirely different domain (infinite sets of real numbers in some bounded
> >interval), while my argument deals with finite sets. Furthermore, your
> >example doesn't make use of the critical inheritance property (everyone
> has
> >exactly one mother) which is the basis for my argument.
> >
> >You can't disprove an argument by presenting an entirely different
> problem
> >and saying that it doesn't lead to the same answer.
> >
> >Thanks for trying, though.
> >
> > Steve Barnard
> >
> >
> The point of the example is to illustrate the fault in your logic. A
> decreasing sequence of numbers may not converge to anything.

There are numbers (real numbers) and then there are NUMBERS (natural numbers).
It is very easy to define an iterated sequence of finite sets with a stochastic
transition rule that converges to a singleton set with probability one. The
sequence I defined is an example.

> A decreasing
> sequence of numbers that is bounded will converge to its greatest lower
> bound which need not be zero. A decreasing sequence of positive whole
> numbers will converge to zero,

The sequence is non-increasing. It converges to a singleton set -- not to a
null set. (Well, eventually it does, at the origin of life!)

> but any finite segment of this sequence
> will not. A non-increasing sequence of positive whole numbers will
> converge to its greatest lower bound, which may not be zero.
> You cannot claim that the size of your set of great .... grand mothers
> reaches one at some point simply because each set (going backward by
> generation) is not larger than the previous one. On logical grounds alone
> the number of females in this "lca founding population" could be any
> number no larger than the smallest known human population size at some
> time.

Are you claiming that the set of mothers will (each) have exactly one daughter
whose lineage continues to the present day, with probability one? And that
this amazing probability will extend back in time forever? That's what's
required for the size of the set to never decrease.

Steve Barnard