Re: Date for Last Common Ancestor?

19 Aug 1996 19:07:30 -0400

In article <>, Stephen Barnard
<> writes:

>CHESSONP wrote:
>> In article <>, Stephen Barnard
>> <> writes:
>> >
>> >CHESSONP wrote:
>> >>
>> >>
>> >> An example. Let I_k be the interval of real numbers greater than
>> 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
>> >entirely different domain (infinite sets of real numbers in some
>> >interval), while my argument deals with finite sets. Furthermore,
>> >example doesn't make use of the critical inheritance property
>> 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
>It is very easy to define an iterated sequence of finite sets with a
>transition rule that converges to a singleton set with probability one.
>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
>> 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
>whose lineage continues to the present day, with probability one? And
>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

Help me flush out the model.

Let H be the set of all human beings alive at time t_0. Let M_0 be
the set of mothers (living or dead) of the members of H. Represent the
functional relationship between H and M_0 by the symbol f, defined by

m=f(h) means that m "is the mother of" h.

The function so defined, f:H -->M_0, is onto, meaning that every element
in M_0 has a "preimage" in the set H. Since H is finite, this implies
logically that M_0 is finite, and, indeed, that the size of M_0 is no
greater than the size of H.

With this functional relationship we can define a sequence of sets
{M_k} for all positive numbers, k, by defining:

"m is an element of M_k, if, and only if, there exists an
element d
of the set M_k-1, such that m=f(d)."

About the sets M_k, we know only that the size of M_k is no greater
than the size of M_k-1. We will need further assumptions to get the
sequences of sizes of {M_k} to converge to one.

I apologize for the clumsiness of the notation, but without a decent
symbol set I think this is the best I can do.