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Re: Date for Last Common Ancestor?
CHESSONP (chessonp@aol.com)
15 Aug 1996 04:47:24 -0400
In article <32105A53.370B@megafauna.com>, Stephen Barnard
<steve@megafauna.com> writes:
>Stephen Barnard wrote:
>>
>> CHESSONP wrote:
>> >
>> > "Let S_1 be the set of all people whom are alive today. Let S_2 be
the
>> > set of mothers of the members of S_1. In general, let S_k be the set
>> > of mothers of S_k-1. The size of these sets is nondecreasing (S_k <=
>> > S_k-1), because everyone has only one mother, but some mothers have
>> > more than one child. When the size of the set reaches 1 then we have
>> > arrived at the mitochondrial Eve.
>> >
>> > Steve Barnard"
>> >
>> > This does not prove that the cardinality of the sequence of sets
{S_k}
>> > coverges to one. To establish the existence of a "Last Common
Ancestor"by
>> > logical means only will require quite a lot more than you have
supplied,
>> > starting with a clear definition of what is the "Last Common
Ancestor"
>for
>> > a species.
>>
>> I suppose that in strict mathematical terms it doesn't. That would
>> require a strictly decreasing cardinality of sets.
>>
>> From a practical point of view I'm satisfied with it. The group that
>> included the putative mitochondrial Eve would very probably have been
>> pretty small. But you never know. :-) I've seen speculation in this
>> newsgroup that relies on *much* flimsier assumptions.
>>
>> Steve Barnard
>
>I've been thinking about this some more, and I'm becoming convinced that
the
>argument for a single mitochondrial Eve, while not airtight, is about as
>close to
>airtight as anything gets in paleoanthropology. Bear with me.
>
>If the cardinality (i.e., the size) of the sets were strictly decreasing
>there
>would be no problem. We would eventually arrive at the singleton set.
>That's
>just simple mathematical induction.
>
>There is no question that the sizes of the sets does not increase. That
>would
>require at least one person to have more than one mother. Obviously,
that's
>not
>possible. I think I'm on pretty safe ground there.
>
>The only problem arises when the sizes of the sets remains unchanged.
>Suppose
>S_k = S_k-1 = N (where I mean the *sizes* of the sets). This would mean
that
>
>every member (i.e., each of the N mothers) in S_k had *exactly* N
daughters
>who
>were in S_k-1, who in turn must all bear female children (because they
are
>mothers of mother in subsequent sets).
>
>Clearly, for large N this is extremely unlikely. But it's worse than
that.
>Even
>for small N (say N=2), for N never to reach one it would be required that
>both
>mothers in set S_k-1 be the *only reproducing daughter* of the two
mothers in
>set
>S_k, and this state of affairs would have to be maintained backward in
time
>*in
>perpetuity*!
>
>So it appears to me that the case for a single mitochondrial Eve is made.
If
>
>anyone can point out a flaw in this argument (assuming anyone has waded
>through
>all this), I'd be very interested to hear it.
>
> Steve Barnard
>
>
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.
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