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#1 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-09 23:37:48

Calligar wrote:

Lol, interesting how you specifically wanted to avoid what Relentless and I found.

I found the point that you and Relentless brought up intriguing and I have engaged with it at considerable length - see my entry at 2016-01-08 17:51:00. But there are two paradoxes here, both of them interesting and each highlighting different aspects of infinite series. If you leave the diary paradox in its original form, the statement that the gods required Achilles to 'eventually take all the pages' allows Achilles to postpone taking non-birthdays indefinitely. This is an interesting observation, but it obscures my original point that the subset of birthday pages is equinumerous with its complement (i.e. countably infinite).

I've been toying with the idea of starting a separate thread to highlight the 'permanent postponement paradox'...what do you think?

#2 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-08 20:43:17

I have now revised the Diary Paradox to avoid the 'Paradox of Permanent Postponement'. The former hinges on the ordering of infinite series whereas the latter highlights the potential pitfalls of conceptualising infinite time.

#3 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-08 20:25:29

Hi Relentless,

I was stuck on the apparent contradiction between statements 4 and 6, but I feel I have now resolved that at least in my own mind. A long-winded way of arriving at the conclusion that you have just concisely summarised!

#4 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-08 18:51:00

I do not feel that we (or at least I) have fully resolved the point that Relentless and Calligar raised above, namely that Achilles can endlessly postpone the task of taking non-birthday pages unless the gods insist that he reveal his page-tearing strategy at the outset and insist that he stick to it. Let’s look at a simpler scenario.

Achilles promises to perform a task but is not specific about when he will perform it. In the case of the Diary Paradox the task could be tearing out the first non-birthday page. But it could be any task – buying his mother flowers, whatever. For simplicity, assume that the task can be accomplished in less than one day.

Recall that Achilles is immortal. Let us assume that he lives in a version of Earth where day and night continue endlessly in their present form. Consider the following statements:

1. Every day is followed by another day, indeed by infinitely many other days.
2. Achilles can postpone the task for n days without breaking his word, for any positive integer n.
3. Ditto, for all positive integers n.
4. Achilles can postpone the task forever without breaking his word.
5. If Achilles postpones the task forever he will never undertake it.
6. If he never undertakes the task, he will break his word.

Where is the flaw in these arguments? One could question whether we can meaningfully posit endless time, but let’s put that aside for the moment.

The flaw lies with Statement 6. The phrase ‘he will break his word’ implies (can only imply) that a time will come when it will be clear that he has broken his word. But no such time will ever arrive. Hence, Achilles can get away with never performing the task without breaking his word. In fact, it is impossible for him to break his word because a time will never arrive in which there is no future in which to perform it, should he so choose. Hence, his promise to perform the task ‘sometime in eternity’ has no meaning.

One could argue that Achilles has broken his word the moment he decides (secretly or otherwise) that he is never going to perform the task. But that is another issue.

If you were fooled even briefly by Statement 6, it is worth considering why. We are able to make conclusions about ‘forever’ based on logic: we have done this in Statements 1-5. We are also able to conceive the completion of infinite series, as we do for example when contemplating the series ½, ¾, 7/8 etc in Zeno’s paradox. So it is fairly easy to fall into the trap of making statements about ‘what happens at the end of infinite time’. If Achilles never performs the task it will be clear that he has broken his promise… at the end of infinite time, which never arrives.

#5 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-06 22:16:19

Calligar, yes, good point. In any finite time interval (is there any other?) a strategy of indefinite postponement is potentially compatible with the ‘take every page’ condition. If the gods were smart they would insist that Achilles declare his page-taking strategy at the outset and stick to it. To fulfil the condition he would have to specify an infinite sequence n1, n2 etc such that on each day n-sub-i he would take a non-birthday page. Achilles is free to space the terms of the sequence as widely as he wishes, eg n1 = 100, n2 = 100^100, n3 = 100^100^100 etc. He might want to take a non-birthday page every now and then just to appreciate how much better off he is. See also my response to Relentless at 2015-12-20 17:19:59.

#6 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-06 15:41:33

The ‘Second Achilles Diary Paradox’ that I mentioned above arises from ambiguous language.

Immortal Achilles is charged with tearing out one page each day from a diary with infinite pages: page 1 on day 1, page 2 on day 2 etc. Consider the following statements:

1. Every page will eventually be torn out.
2. All the pages will eventually be torn out.
3. All the pages will eventually have been torn out.

The first statement is true because Page n will be torn out on Day n for all n.
The third statement is false because there is no n such that on Day n, every page will have been torn out.
The second statement is potentially ambiguous: true if interpreted as meaning statement 1, false if interpreted as meaning statement 2.

Let us give Achilles a new assignment. He is required to walk a distance of 1 kilometre in a straight line from a point A to a point B. He must first cover half the distance, then half of the remaining distance, then again half of the remaining distance, and so on ad infinitum. This is one of Zeno’s paradoxes. Let us assume he walks at constant speed. Consider the following statements:

1. For every positive integer n, Achilles will eventually reach the point ½ + ¼ + 1/8 +…2^(-n) kilometres from A.
2. For all positive integers n, Achilles will eventually reach the point ½ + ¼ + 1/8 +…2^(-n) kilometres from A.
3. Achilles will eventually have reached all the points ½ + ¼ + 1/8 +…1/2^n kilometres from A.

Again statement 2 is ambiguous in that it could be taken as equivalent to statement 1 or statement 3. I assume Zeno had no difficulty accepting statement 1 but he balked at statement 3 because it implied the completion of an infinite sequence.

The funny thing is that none of the statements is true because, as noted earlier in this thread, space cannot be infinitely divided.

Discuss!

I’ll try to find time soon to respond to the recent comments above.

#7 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-04 12:35:18

I would like to add a comment on a point that I touched on earlier. It concerns the condition that the gods place on Achilles to ‘eventually take all the pages’. This is (to use Bob Bundy’s word) slippery.

Any given page in the diary will be torn out in finite time - no page will forever remain untorn - so in that sense Achilles will fulfil the condition. Let us call this Conclusion 1.Yet a day will never arrive when Achilles has taken all the pages, so in that sense he will never fulfil the condition (Conclusion 2). This in itself is a paradox, quite independent of the order in which he tears out the pages.

How to resolve the paradox? We could argue that the gods’ condition is ambiguous, but I don’t think this is the whole story. The main point is that unlike a converging sequence of intervals of time (or space), eg ½, ¾, 7/8... seconds, an infinite sequence of equal time intervals cannot be completed – at least not as far as we know, from the perspective of daily life. Mathematically we might say that Achilles' days can be numbered by the integers, but not by the integers plus omega (the first transfinite ordinal).

#8 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-03 23:52:35

Hi Relentless, thanks for the welcome!

If you have two boxes A and B, and B fits inside A but not vice versa, it seems rather difficult not to conclude that A is larger than B, regardless of whether the universe is finite or infinite. Whether or not the boxes are infinitesimal compared to the universe has no bearing on the comparison or on the notion of comparative size. If there are folks out there who question this, they sound rather interesting and I would like to meet them! Of course if the two boxes are at opposite ends of the universe, or even a few kilometres apart, the comparison becomes a little more tenuous.

I find the notion of probability persuasive yet at the same time somehow dubious. It relies on the assumption that an event can be repeated, but one of the odd things about life is that nothing is repeated; what is is, and what was is only memory. This strikes me as one of the greatest mysteries of existence, yet for the most part we blithely disregard it. Repetition implies time and time is…?

I wholeheartedly agree that there is almost nothing about the world we can be absolutely sure of. It is by no means clear that there is such a thing as ‘the world’ other than the sum total of our beliefs, impressions and expectations, which one would be rash to assume are entirely coherent. Descartes made a brave attempt to question everything, but he foundered badly when he assumed that there was an ‘I’ who was doing the thinking – apparently it didn’t occur to him that the ‘I’ might itself be a figment of thought.

We need to consider the possibility that thinking, being an exercise in virtual reality, cannot open the window to reality (if there is such a thing), although it can investigate its own processes (such as preconceptions) that interfere with clear perception. I am inclined think the Vedic seers and their ilk were on the right track when they advocated a state of mind in which the movement of thought is in abeyance.

#9 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-02 23:19:51

Relentless - In response to your post about measurement: I do not feel qualified to discuss the size of the universe. As for its divisibility, I agree with your point that continuity (of space for example) would seem to be unverifiable. I have often wondered for example whether it makes any sense to consider (say) a space interval of 10 to the minus 100 metres, given that there is no known way even in principle of dividing 10 to the minus 99 metres into so many ‘equal intervals’.

You suggest there is a difference between whether something is essentially determined and whether it is predictable. If something were essentially determined but not predictable, how would we know? And if we could not know, on what basis could we claim that it was essentially determined? I feel that this question could potentially cast light on a deeper question concerning the relationship between ‘the world as seen through the prism of measurement’ and ‘the world as it actually is’. Put another way, is there an actuality/reality beyond the realm of measurement, and is it possible to perceive it?

#10 Re: Puzzles and Games » The Achilles diary paradox » 2016-01-02 22:53:37

Hello Calligar and Relentless,

My apologies for being ‘ambulando’ for so long, but I appreciate your discussion. Relentless, you have clearly grasped the essence and details of the paradox, and your last sentence (‘The principle is that he can choose the order of an infinite set…’) summarises the essential point concisely. The gods are of course irrelevant and added purely for the sake of spinning a yarn!

You suggested reworking the paradox so as to abolish the diary, suggesting that ‘[Achilles] has the ability to choose which order the days of his life will take place’. I considered presenting the paradox in this way, but was reluctant to do so because it introduced the additional complication of seeming to require time travel or the rearrangement of time.  I felt that this complication (adding science fiction to fantasy!) would obscure the main point of the paradox and make it harder for people, or harder for some people, to see the essential point which is concerned not with reordering time per se but with reordering an infinite set. Hence the diary.

Calligar, in response to your question: Each page that he tears out determines his fate for that day. Sorry, I can see now that ‘tearing out’ is potentially ambiguous! I envisaged tearing the pages as one might tear off the pages of an office daily planner – ‘Tuesday: Meeting with Sales Team’ etc.

I have now edited my original entry – I hope it is a little clearer.

#11 Re: Puzzles and Games » The Achilles diary paradox » 2015-12-20 20:46:31

That’s the kind of trouble I’d gladly get into! I find these questions fascinating not only because they challenge (and perhaps exceed) the intellect, but also because I suspect they contain deep clues regarding our understanding, and indeed our misunderstanding, of space and time and hence of our lives generally.

For example it is fact that our powers of measurement are limited in practice, and physics suggests that they may be limited in principle; yet we persist in modelling physical space with a number system (the reals) that is not only infinite but uncountably so. It seems to me that Zeno was amongst other things attempting to draw attention to the apparent disjunction, indeed irreconcilability, between points (numbers) and extensions (intervals). Set theory has gone some way to resolving this matter mathematically, yet the continuum hypothesis remains unproven and the point/extension problem remains unresolved in terms of its application to physical reality. Indeed I suspect (I’m sticking my neck out here, being a rank amateur) that this question could be relevant to if not lie at the heart of the apparent irreconcilability of relativity and quantum theory.

I suspect we need to rethink our whole notion of points and intervals as an adequate model of our physical and temporal environment. Indeed I think the whole notion of measure could do with a thorough raking-over.

As for choosing to die, it would not necessarily be morbid if we could perceive deeply (as perhaps various ‘seers’ have over the ages) the significance of transience (time), and so in some sense be free of it. Your comments raise deep questions about the nature of self, identity, experience, and much else besides.

I’d be interested in exploring these questions further, with the proviso that I will not be able to spare much time and will often be ‘off the grid’ for extended intervals (the ‘ambulando’ part of my moniker). For starters, I’d be interested to learn why philosophically you find it more probable that space is quantised.

Would you be interested in starting a new thread in the ‘Dark discussions’ section?

#12 Re: Jokes » Einstein's formula » 2015-12-20 08:39:56

I figured it out but I can't remember how exactly. I thought about it for some time and considered the various ways in which the formula could be interpreted.

More info

#13 Re: Puzzles and Games » The Achilles diary paradox » 2015-12-19 22:14:13

Yes, infinity is fun and also intriguing because as Zeno pointed out it seems to be unavoidable – you can’t walk across a room without stepping over it!

Achilles can delay the non-birthdays by specifying a mind-boggling large number, but from the perspective of infinity any finite span of time is but the blink of an eye. As you say, he has to pay his dues eventually.

Perhaps the real horror of hell, if there is such a place, is not that it is a place of suffering but that residence therein is endless. One could say the same of heaven! It is strange, is it not, that the prospect of dying is less frightening than the prospect of living forever? These musing probably belong in the ‘Dark discussions’ section.

#14 Re: Puzzles and Games » The paradox of the infinite pyramid scheme » 2015-12-19 18:35:21

Pyramid schemes can come unstuck, and if your room number exceeds Graham’s number you could be waiting a long time to join the Collective, let alone get rich. Fortunately there’s a faster and morally respectable way for everyone to join the gravy train, namely banking!

Everyone in Hilbert’s Hotel pays $1 into a bank. The bank then redistributes the contributions from Rooms 1 up to N to the occupant of Room 1, the contributions from Rooms N+1 to 2N to the occupant of Room 2, and so on. That way everyone gets $N, and N can be set arbitrarily large. Isn’t banking a wonderful thing.

#15 Re: Puzzles and Games » The Achilles diary paradox » 2015-12-19 18:19:59

Good point Relentless. You’ve stumbled upon Part 2 of the paradox! Whenever the gods say ‘Achilles, isn’t it time you took your first non-birthday page?’, Achilles can always reply ‘Not yet!’ without violating the conditions of his immortality. But if he *always* replies ‘Not yet!’ he will violate those terms!

How to resolve this apparent contradiction?

The answer (I think) is that the two statements relate to two different perspectives on infinity. From the perspective of the here-and-now the future is indeterminate so the first non-birthday can be delayed to some unspecified future. From a mathematical perspective, if the conditions are to be satisfied there must be some numbers n1, n2 etc such that on Days n1, n2... Achilles takes non-birthday pages. If the gods were smart they would insist that Achilles specify his page-tearing strategy in advance: that is, specify the days on which he would choose non-birthdays.

I think Zeno was driving at a similar problem with his paradox of Achilles getting from A to B. From the perspective of the halfway mark , three-quarters mark etc, Achilles always has further to go; but from the vantage of point B he has arrived. The difference is that in the spatial case the infinite ‘steps’ can be completed, whereas in the case of endlessly succeeding days they can’t – not even if one is immortal!

#16 Re: Jokes » Einstein's formula » 2015-12-19 10:28:24

My answer

Monox, you were getting warm!

#17 Re: Puzzles and Games » Can anyone help me with this riddle? » 2015-12-08 06:37:30

Seems any words that look vaguely rude are automatically replaced with 'math'. Fair enough to keep the site clean, but too bad if you want to write about roosters.

#18 Re: Puzzles and Games » Can anyone help me with this riddle? » 2015-12-08 06:28:21

Something really weird going on here. WEATHER  C O C K E R E L. Has the spell checker gone crazy?

#19 Re: Puzzles and Games » Can anyone help me with this riddle? » 2015-12-08 06:26:18

Huh??? I meant to write weather math. Too early in the morning.

#20 Re: Puzzles and Games » Can anyone help me with this riddle? » 2015-12-08 06:25:03

My guess is a weather math. The second head is the head of the arrow. See for example

http://www.dreamstime.com/royalty-free-stock-images-rooster-weather-vane-weathercock-silhouette-image32290899

#21 Re: Puzzles and Games » The paradox of the infinite pyramid scheme » 2015-12-07 21:28:29

As you're probably aware the phrase also refers to Diogenes of Sinope's response to the assertion (implied by Zeno's paradox) that motion is impossible - he got up and walked! Personally I'm impressed and infuriated by this response in equal measure.

I'm glad I'm not the only one who says dumb things in posts! Fortunately, nothing that a bit of perambulation can't rectify.

#22 Re: Puzzles and Games » The paradox of the infinite pyramid scheme » 2015-12-07 18:51:45

Thanks Bob. The new name suits me better, as my passion for walking is even greater than my passion for mathematics.

#24 Re: Puzzles and Games » The paradox of the infinite pyramid scheme » 2015-12-07 10:06:51

Hi Bob,

It's my turn to say Hmmmm! I hadn't taken account of the prevalence of misspelling. Yes Zeno-phobia strictly means fear of Zeno...but Zeno was a peddler of paradoxes about infinity. It's a weak pun and in the current climate of escalating xenophobia, probably ill-advised. Is it too late to change my user name? Perhaps I should just de-register and sign up again?

Cheers,
Z

#25 Re: Puzzles and Games » The paradox of the infinite pyramid scheme » 2015-12-06 23:52:20

Hi Bob,

Thanks for your welcome. Zenophobia translates roughly as ‘Fear of infinity’! (A pun on Zeno obviously.) A much healthier fear than the X-variety, imho. I’m not sure if that counts as ‘fully understanding’ my moniker.

Yes, the problem with Capitalists is not so much their fear of infinity but their inability to understand the implications of finity.

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