Thanks for the many replies to Challenge #65! I include the original
problem, followed by a representative sample of thoughtful responses
which are published below.
Keep supporting Zeno's, which is totally for critical thinking and is
never going commercial!!!
of the Possible?
A couple of
Zeno's patrons were talking the
other night about what is, and what is not, possible (it was late, and
expresso special of the evening was quite robust). Their line of
indeed in keeping with Zenoesque dialogue, it seemed, so we include it
introduce our latest Zeno's Challenge...back by popular demand. But you
be the judge as you pull
chair, and please offer your response at the end of the scenario---and
to have it
posted later at the Coffeehouse, if you so choose to reply, as I hope
you shall....Ron Barnette
were asked to write down
very quickly the number '1'---say, within ten seconds," posed Maggie to
her evening companion.
"No problem," retorted Charles, smirking.
"Ok," Maggie added, "And further suppose that you were asked to
write down the number '2' within half the time it took you to write
previous number---could you do that?"
"Of course," quipped Charles. "No problem. What's the big
"And how about the number '3' within half the time it took to jot down
'2', and the same with '4' in half the time taken to write down
manage that?," pushed Maggie.
"Yes, I think so," said a confident Charles, who grabbed a napkin and
quickly scribbled down the four numbers well within the eighteen
seconds or so
time frame prescribed by the rule. "Here you are," he gestured.
"Well, Charles, I've been wondering about what would be the logical
of an all-powerful being," added Maggie. "I mean, an omnipotent God,
or any other super computational device not limited by physical
Just how far could such an entity proceed with our little game? For
could such a being continue to produce the next hundred numbers in like
fashion, with each written in one-half the time taken to write the
one," she queried.
"Wow, that would be really fast," Charles responded, "But an
all-powerful being could do it, I believe."
"Let me pose my main question, Charles," Maggie summarized.
"Suppose that we use as a concrete time frame our one-hour visit here
tonight at Zeno's, from 11pm to midnight, and consider the following
this framework. First, could an all-powerful being follow the rule of
down the next number in the series in half the time taken to write down
previous number, beginning as you did with '1', to be written within
but do this for ALL the natural numbers---in other words, by not
running out of
time in following the rule, as you and I would? And second, if an
being could successfully follow the rule---think of it like RUNNING A
PROGRAM---wouldn't it produce all the natural numbers, and there are
many, well within the hour's time we spent here tonight?"
"Now that's too weird!," remarked Charles. "But let me think
about what would be involved---maybe in the morning---hmmm...."
are samples of the many replies to this Zeno's Challenge. Many
thoughts of appreciation are extended to those who continue to support
and drop by Zeno's!!! Keep up your support as you submit your
and discussions! Thank you much...Ron Barnette
Topi in Finland:
From Rob Schmidt in
It's been a while since I've stopped by Zeno's, but glad to
see you are still in business! Here's some thoughts on the
"Limits of the Possible?" puzzler.
Mathematically, I believe the answer to how long it
would take to count all the natural numbers given the time constraints
is "20 seconds". I can't write the proper mathematical
symbols here, but it's the following sum: 10 seconds + 1/2(10
seconds) + 1/4(10 seconds) + ... + 1/2^n(10 seconds). (1/2^n
is "one over 2 to the power of n.) Factor out the
common 10 to get the sum = (10 seconds)x(1 + 1/2 + 1/4 + ... +
1/2^n). The sum of the fractional portions as n approaches
infinity is well-known, and equals 1. So, the total time is
(10 seconds)x(1+1) = 20 seconds. (Assuming I did my math
Now, for reality. According to modern
quantum mechanics, time is quantized, and the shortest possible time
period (known as the Planck time) is about 5.39 x 10^(-44) seconds
(that's "times ten to the -44 power"). At that point, any
being in our universe, omnipotent or not, could not (in theory) split
time any finer, and so the counting would stop somewhat shy of
infinity. It might be an interesting exercise to calculate
what would be the value of "n" (i.e., how
high the omnipotent being would have
counted) when the time interval finally reaches the Planck
time. As in my old freshman physics books, this is left as an
exercise for the reader. ;^)
Funny you should ask as this is just Zeno's paradox revisited.
So if writing down 1 takes 10s, 2 5s, 3 2,5s etc. the whole thing, from
a entity that could do it, takes only 20s.
From Steven Splinter
Maggie's experiment works,
because the all powerful being will approach
but never reach the alloted time span. It will always go half-way of
the last travelled distance infinitely, and so compute every natural
number in the process. Plus, since the initial time alloted is 10
seconds instead of 1 hour, the time span the being is working in is 10
seconds, leaving 59:50 worth of time never traveled by the being in its
From Nikolaj Nottelmann in Denmark
Maggie's challenge is a variation of the half-way journey thought
experiment. In it, a man with a certain goal at a set distance goes
halfway to that goal, then halfway of the distance he just traveled,
then halfway of the previous distance, ad infinitum. It becomes an
infinite trek because the goal is never reached.
In Maggie's challenge, time is the distance that must be traveled. It
doesn't matter if it is one hour's worth of time, or half an hour, or
even 1 second. All that matters is the being in question must be able
to constantly and consistently travel half the distance just previously
traveled. The initial goal, be it 1 mile or 1 minute, serves to define
what will be the initial whole value that the being must travel half
of. From there on, the distance just traveled becomes the whole that
the being must now travel half of, and so on, repeating infinitely.
The distances traveled will add up over time. The first value will be
.5 of the whole; the second value will be .25 (which is .5 of the .5
traveled in the previous step), making the total distance traveled .75
of the whole. The values add up, but never amount a whole 1 - there
will always be some missing amount, however infinitesimal. Practically,
we may say that the number comes close enough for the purposes of
whatever the current calculation is for, or in respect to the practical
limits of the technology used, similar to how we define significant
figures in rounding calculations. Maggie's experiment doesn't need to
work around these limitations, though, partly because she has a
measuring device in the form of an all powerful being that is
presumably infinite in it's measuring and computational abilities.
She's also set up her experiment so that it can't help but calculate
all of the numbers in the time allotted.
Note that Maggie defines the first whole value to be traversed not as
one hour, but as 10 seconds. This means the amount of time the program
runs will always increment towards, but never reach, 10 seconds. The
first step will be 5 seconds, the next 2.5 seconds, and so on. The
other 59:50 of time in the hour are irrelevant. The hour is never
reached! Maggie and Charles can spend the rest of the hour doing
whatever they want to do - hopefully something more interesting than
watching an all powerful being calculate numbers. I know I can think of
a few things I'd rather be doing!
If nothing else, let this be a lesson never to do anything half-way!
Dear Ron Barnette.
Great fun! Could not help posting a response. Hope it is in the right
Response to Zeno Challenge "Limits of the possible".
1. The nature of the problem.
the intended nature of the problem is this: Is it logically possibly
that all natural numbers be inscribed subsequently within a limited
period of time, using a programme cutting the inscription time for each
subsequent inscription in half? E.g. the numeral "1" must be inscribed
within [0, t[ "2" must thereafter be written down within [t,t+t/2[ etc.
The constraints on this reading are:
1 must be the first numeral inscribed.
numeral other than 1 may be inscribed within [0;
n must be inscribed within [t/20+t/21+....+t/2n-2 ;
numeral except n may be inscribed within [t/20+t/2+....+t/2n-2 ;
there is an alternative reading not obviously ruled out by the problem
text. According to this reading the constraints are somewhat simpler:
A*. No numeral may be inscribed before 1.
B*. Numeral n must be inscribed before t/20+t/2+....+t/2n-2
C*. Numeral n+1 must not be inscribed until numeral n is inscribed.
illustrate, a programme will have complied with A*-C*, but not A-E, if
it at once writes down all natural number numerals in one fell swoop.
This at least complies with one reading of
"could an all-powerful being follow the rule of writing down the next
series in half the time taken to write
down the previous number, beginning as you
with '1', to be
written within ten seconds, but do this for ALL the natural numbers",
t=5 seconds. I shall however go with the stricter requirements A-D, as
those seem more loyal to the general spirit of the problem. Remark that
having complied with A-D or A*-C* logically implies having complied
with the overriding requirement:
E. All (natural number) numerals must have been inscribed within [0; 2t]
since [0; 2t] is a limited time-interval, that may be set at any lenght
according to specifications.
2. Putting aside misplaced objections.
challenge asks for a program "writing down" numerals, hence clearly
implies that some mechanical form of reproduction be involved. Problems
of limited ink and paper supplies may be overcome by recycling, since
the text does not require for all numerals to be ever inscribed at one
time. It is for this reason that I have not put E as the stricter
requirement that All natural numbers must be inscribed before
eventually the device inscribing the numerals must break the speed of
light, as it accelerates in order to complete its assigment. This
violation of the laws of nature, however, is deemed irrelevant by the
3. Any real problems?
In order to work,
the programme must be implemented and its implementation must causally
interact with a mechanism of printing in order to produce numeral
entokenings. But since normal laws of physics do not apply, the
implementation can presumably be within God's mind with God then
outputting inscriptions onto some medium through Divine Intervention.
And since it follows that, if God adheres to A-D, he adheres to E, the
only problem is then whether God can adhere to A-D. But A-D involve no
problematic omnipotence-threatening demands like creating a stone one
cannot lift. So since God is presumably omnipotent, he can adhere to
A-D: No time-span is too short for an omnipotent being in order to
divinely intervene and complete a task. Hence he can adhere to E and
complete the desired task.
4. Problems stemming from causal concepts.
"easy" solution above required for a crucial premise: The
possibility of causal intervention in defiance of physical laws (even
divine intervention is a causal link, since
<intervention> is a
causal concept). But if one prefers a mechanistic analysis of causal
concepts, it becomes a conceptual necessity that a mechanism was
involved in any intervention. And a mechanism may then be
an exchange of physical magnitudes like energy, momentum etc. It would
then seem that even God is conceptually required to adhere to basic
physical laws in so far as he aims to intervene in the physical
universe and print numerals. However, this proposed solution crumples
once it is recognised that God's omnipotence must surely include the
power to change the laws of nature. Hence, before each subsequent step
of complying with C, God can simply change any law of nature standing
in the way of divine intervention in order for him to subsequently
comply with the laws of nature currently in operation. If he cannot do
this he is not truly omnipotent. But the problem did not allow for
questioning the possibility of a truly omnipotent being.
argued above I see no principled reason why it is not logically
possible for an omnipotent being to write down all natural number
numerals in accordance with the rules A-D (hence E) short of arguing
that a truly omnipotent being is a contradiction in terms or that
logical possibility is really a species of physical possibility.
However, arguments to that conclusion would clearly violate the terms
of the problem text.