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!!!
Ron Barnette

Limits 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 the expresso special of the evening was quite robust). Their line of thought was indeed in keeping with Zenoesque dialogue, it seemed, so we include it here to introduce our latest Zeno's Challenge...back by popular demand. But you be the judge as you pull up a 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

"Suppose you 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 down the previous number---could you do that?"
"Of course," quipped Charles. "No problem. What's the big deal?"
"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 '3'---could you 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 limits of an all-powerful being," added Maggie. "I mean, an omnipotent God, or any other super computational device not limited by physical constraints. Just how far could such an entity proceed with our little game? For instance, 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 previous 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 within this framework. First, could an all-powerful being follow the rule of writing down the next number in the series in half the time taken to write down the previous number, beginning as you did with '1', to be written within ten seconds, 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 all-powerful being could successfully follow the rule---think of it like RUNNING A PROGRAM---wouldn't it produce all the natural numbers, and there are INFINITELY 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...."

 Here 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 replies and discussions! Thank you much...Ron Barnette

From Rob Schmidt in California:

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 correctly.)

 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.  ;^)

From Topi in Finland:

 Funny you should ask as this is just Zeno's paradox revisited.

sum(i=0,oo)(2^-i)=2

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

Briefly: 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 calculations.

Explicated:

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!

From Nikolaj Nottelmann in Denmark
Dear Ron Barnette.
 
Great fun! Could not help posting a response. Hope it is in the right mould!
 
Response to Zeno Challenge "Limits of the possible".

1.    The nature of the problem.

Clearly 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:
    A.    Numeral 1 must be the first numeral inscribed.
    B.    No numeral other than 1 may be inscribed within [0; t[.   
    C.    Numeral n must be inscribed within [t/20+t/21+....+t/2n-2 ; t/20+t/2+....+t/2n-1 [
    D.    No numeral except n may be inscribed within [t/20+t/2+....+t/2n-2 ; t/20+t/2+....+t/2n-1 [

 However, 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.

 To 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 number in the                  series in half the time taken to write down the previous number, beginning as you did        with '1', to be written within ten seconds, but do this for ALL the natural numbers",
setting 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.

The 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 2t.  But 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 problem text.

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.

The "easy" solution above  required for a crucial premise: The logical 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 construed as 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.

5.    Conclusion

As 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.