| HPS 0628 | Paradox | |
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For submission
1. The Hilbert hotel is half full. All even rooms only are occupied by guests. The hotel owners are pressing the manager to find more guests to fill the empty rooms, but all his efforts to advertise for new guests fail. What can the manager do to fill the hotel and keep his job?
2. The manager of the Hilbert hotel can arrange for there to be more empty rooms in his full hotel by moving guests among the rooms. Each scheme we have seen requires him to move infinitely many guests. That produces infinitely many annoyances. Is there some way he can move the guests among the rooms to make more space while moving only finitely many guests?
3. In a pyramid marketing scheme, the single first tier agent recruits 10 second tier agents. Each agent pays the first tier agent $100, so the first tier agent makes $1,000 in profit. The second tier agents now each recruit 10 third tier agents under the same scheme and each are paid $1,000 by the third tier agents. Since each second tier agent paid the first tier agent $100, each second tier agent makes a net $900 in profit. The scheme continues through as many tiers as can be marketed.
a. If
the number of agents grows as:
1 agent total in the first tier.
11 agents total in the first and second tier.
111 agents total in the first, second and third tier.
1,111 agents total in the first to fourth tier.
etc.
How many tiers would it take for the total number of agents to exceed the
total population of the US (roughly 330 million)?
b. What happens to agents' profits when no new agents can be recruited?
c. What would happen if the scheme were implemented in a country with infinitely many people?
Not for submission
A. Examples such as these fueled suspicion of infinities for a long time. Are these paradoxes sufficient in the end to preclude infinities from serious analysis?
B. Would there be any day in Shandy's life that is not included in his autobiography?