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...."

 Please share your thoughts!!! Thank you, as your ideas will be published for others to read...

For your ongoing enjoyment, I have retained these so-called Impossible Objects on here, as you contemplate the new challenge. I continually receive positive comments from many loyal Coffeehouse patrons. If you locate other good ones, please let me know. Please do this for our many visitors. Enjoy!...Ron Barnette

Some Impossible Objects

I credit Jim Loy for these marvelous example of so-called impossible objects, which should amuse you faithful Zeno's patrons, who appreciate thought-provoking stimulation.

		1. Asymmetric crate. An M. C. Escher creation, I surmise
		2. The Penrose staircase: Often drawn by M. C. Escher. Clockwise is downstairs forever.
		3. The tribar: Another impossible object by R. Penrose. It's hard to know how to color it, as the interior becomes the exterior.
		4. The Penrose triangle: Another famous impossible object by R. Penrose. This is sometimes called a tribar (see #3, above).
		5. An ambiguous ring...What is the outside? The inside?