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Here is another paradox, this time based on Hilbert's paradox of the Grand Hotel.
The Hilbert Hotel has infinite rooms numbered 1, 2, 3 and so on indefinitely. Each room has exactly one occupant and each occupant initially has $100. Infinitely many occupants are of sound mind - sound enough to recognise a good deal when they see one.
Two occupants establish a Collective of which they are the founding members. They then invite two other occupants to join the Collective. The conditions of joining are that one must pay a joining fee of $100 and persuade two other occupants of the hotel who are not yet members to join. Membership fees are divided equally between pre-existing members.
When the first two new members join they pay $100 each, so the original members each receive $100. The membership now stands at four. The new members persuade four new members to join, so the four old members receive another $100 each and the membership increases to eight. The four new members persuade eight new members to join, so again the existing members receive $100. The process continues indefinitely.
However high one’s room number, it will always be possible to join the Collective. However large the Collective becomes there are always infinitely many potential new members. However large the Collective is when one joins it, joining guarantees an income of $100 every time the Collective doubles. Given the incentive of ever-increasing wealth, the Collective will keep doubling indefinitely. Hence everyone in the Hotel can join the Collective and everyone in the Hotel can become fabulously wealthy.
I would like to offer the following ‘semi-paradox’ for discussion.
“The gods tell Achilles that they have some good news and some bad news. The good news is that he will live an infinite number of days. They have even prepared a diary (with infinite pages) foretelling the fortunes that will befall him on each day. Each day when he wakes up Achilles must tear out a page of the diary, and the contents of the page will determine his fate for the day. Every page that carries the date of his birthday states, 'Today you will enjoy divine powers'. The bad news is that due to a printing error, all the remaining pages state, 'Sorry, today is basically just another day at the office'.
As compensation for the error the gods allow him to tear out the pages in any order he chooses, providing (1) he states in advance his strategy for tearing out the pages, and sticks to it, (2) the strategy involves tearing out one page each day, and (3) every page is assigned a specific day on which it will be torn out. For the sake of clarity let us say that his birthday is June 1st and that the first page of the diary is Jan 1st 0001 AD. The diary covers the years 0001, 0002 and so on indefinitely.
Achilles chooses as follows: For the first 364 days he tears out the first 364 June 1st pages; then he takes the first non-birthday page (Jan 1st of the year 0001); then he takes the next 364 June 1st pages; then the second non-birthday page (Jan 2); and so on endlessly, treating himself to an extra birthday on leap years. Thus Achilles enjoys divine powers and celebrates his birthday on all but one day of each year, and will do forever.”
Achilles’ fate seems paradoxical because it is tempting to think that his future birthdays are far less numerous than his non-birthdays and that his selection regime somehow contravenes this. The temptation is reinforced by the fact that in any finite diary his future birthdays certainly will be far less numerous than his non-birthdays; indeed the ratio of non-birthdays to birthdays converges to 364.25 (allowing for leap years). But in the case of infinite time his birthdays are neither less nor more numerous than his non-birthdays; both are countably infinite (there are infinitely many June 1sts) and the ratio of his non-birthdays to his birthdays is undefined.
If Achilles’ lifespan were finite he would eventually run out of ‘birthday’ pages and be obliged to use up a huge backlog of non-birthday pages. For example, if he lived to 100 and if the diary covered 100 years, he would run out of June 1st pages after 100 days and would have to live the remaining 99 years and 265 days without divine powers and without a birthday.
For me the appeal of the paradox lies in the nagging sense that his strategy ought to suffer a similar failure if he is immortal, even though the mathematics demonstrates otherwise.