| HPS 0628 | Paradox | |
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Paradoxes of the Infinitely Large
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton
Supertasks produce problems when we consider the results of completing an infinity of actions. We can go one step deeper into infinity and still find paradoxes. They arise when we simply consider an infinite collection of anything. We shall see in the next chapter that a resolution of the paradoxes lies at the foundation of modern set theory. There we will find a fascinating and unexpected account of infinitely large collections.
The contemplation of an infinity has produced a wide range of paradoxes. On closer inspection, many turn out to be clever retellings of just one paradox. The classic version in the literature is the "Hilbert Hotel." We are to imagine a hotel with infinitely many rooms. They are all occupied.
All is well and the hotel manager is happy. But
then...
A new guest arrives and seeks a room. What can the manager do? All the rooms are full!
The inventive hotel manager has a scheme for making space. He asks each of the guests in the presently occupied rooms to move up one room.
The guest in room 1 moves to room 2;
the guest in room 2 moves to room 3;
and so on for all the guests.
Room 1 is now empty and available for the new guest.
Next, a bus of tourists arrives with 10 guests, each seeking a room. The manager can find space for them as follows:
The guest in room 1
moves to room 11;
the guest in room 2 moves to room 12;
the guest in room 3 moves to room 13;
and so on for all the guests.
Rooms 1 to 10 are now empty for the new guests.
Next door to the Hilbert hotel is the Bernays hotel, also with infinitely many rooms, all filled. A fire alarm forces all these infinitely many guests to evacuate the hotel and seek rooms next door in the Hilbert hotel. Once again the manager can find space for them as follows.
The guest in room 1 moves to room 2;
the guest in room 2 moves to room 4;
the guest in room 3 moves to room 6;
and so on for all the guests.
The infinitely many odd numbered rooms, 1, 3, 5, 7, ... are now empty and available for the infinitely many new guests.
The capacity of the Hilbert hotel to accommodate new guests has scarcely been touched. The manager can move the guests around so that there is always an occupied room surrounded by any number of empty room, no matter how large. For example, the manager might move the guests around as follows:
The guest in room 1 moves to room 101
= 10;
the guest in room 2 moves to room 102
= 100;
the guest in room 3 moves to room 103 = 1000;
and so on for all the guests.
There are now stretches of empty rooms of arbitrarily great length.
Imagine a new guest arrives who demands absolute quiet. If that guest demands at least 100 empty rooms on either side of the guest's room, the manager can put this troublesome guest in room 500. The nearest rooms are room 100 and room 1000. It does not matter how isolated the new guest would like to be, the manager can always accommodate the new guest by moving the guest to a room with a high enough number.
We can now return to the paradox of the expanding universe described in earlier chapters. In modern big bang cosmology the collection of galaxies forming the matter of the universe is everywhere expanding and thus everywhere diluting. How can we have such a dilution everywhere if matter is conserved? Should not a dilution in one part of space be accompanied by a concentration of matter in another?
The earlier chapter sketched one solution. In relativistic cosmologies, space itself is expanding. That is, more space is created to fill the gaps between the galaxies. This happens everywhere and thus the matter of the universe is everywhere diluted.
However this same universal dilution can also arise in non-relativistic, Newtonian cosmologies. In them, the matter distributed in the galaxies can still expand and dilute everywhere. Space remains absolute and unaffected. The expansion of the galaxies is simply a motion of recession within a fixed absolute space.
The mechanism that enables this expansion everywhere is the one just seen in the Hilbert hotel. It depends essentially on space being infinite. A simple way to see it is to look at just one dimension of space in an expanding Newtonian cosmology. For simplicity, we will divide space along this one dimension into huge blocks. Each contains one galaxy. The uniform, universal motion of expansion leads the galaxies to spread apart. Intervening blocks of space are vacated in just the same way as the moving of guests in the Hilbert hotel leads to empty rooms between them throughout the hotel.
This same mechanism can manifest as a creative process. To see it, consider what might happen in the modern world of electronic banking if we could somehow open up an infinity of bank accounts. Let us number them 1, 2, 3, ... In this banking system, we are allowed a small, temporary overdraft on each account, as long as the overdraft is settled soon after it is taken. That is, each account can stray into a negative (red) balance, as long as we soon repay the overdraft amount and return the account to a zero balance or better.
Now consider the following infinite set of transactions. We set them up electronically and ask the bank's computer system to execute them all at once:
$1 is deposited into account 1, supplied by taking
an overdraft of $1 on account 2.
Account 2 is replenished by a $1 deposit, supplied by taking at overdraft
of $1 on account 3.
Account 3 is replenished by a $1 deposit, supplied by taking at overdraft
of $1 on account 4.
Account 4 is replenished by a $1 deposit, supplied by taking at overdraft
of $1 on account 5.
Account 5 is replenished by a $1 deposit, supplied by taking at overdraft
of $1 on account 6.
And so on for all infinity of the accounts.
The overall effect is that we start with nothing in any of the accounts. At the end of the process, we have a net gain of $1. If we are greedy, we could make the same transaction set many times; or just make the transaction once with, say, $1000.
In the banker's paradox, we create money from nothing. This creation from nothing mechanism is the same one that played out in the reversed slow-speed supertask of the earlier chapter. We started with an infinity of bodies all at rest. None has any momentum or any kinetic energy. At the end of the supertask, the first body has acquired some kinetic energy and momentum. All the other bodies are at rest and have neither.
Analogously to the banker's paradox:
This first body borrowed
momentum and kinetic energy from the second body.
The second body borrowed momentum and kinetic energy from the third body.
The third body borrowed momentum and kinetic energy from the fourth body.
And no on for all infinity of the bodies
The net effect is that first ejected body gains energy, while all the remaining bodies that supplied energy are restored to their zero energy condition by an energy contribution from the next body in the sequence.
If the first body gains energy E, this figure shows more clearly the analogy to the banker's paradox:
In both cases, the infinity of the bank accounts and the infinity of bodies is essential. If there were only finitely many of each, then there would be a last bank account or a last body in the sequence of transfers. In the banker's paradox, there would be no further accounts to remove the overdraft. Any gain made in one account would be offset by an enduring overdraft in others. Similarly, in the slow-speed supertask, there would be no further body to supply the needed momentum and kinetic energy. Any gain in energy or positive momentum by the ejected body would be offset by a negative energy and negative momentum in other bodies.
Here's how things might end in the case of finitely many colliding bodies:
The ejected body could accrue energy E,
only if some other bodies could somehow adopt a negative energy state. In
the figure, the last body in the finite set of four bodies adopts energy
state -E.
There is no violation of conservation of energy since
Total energy at start = 0
Total energy at end = E
+ -E
= 0.
---oOo---
These examples of the paradoxes of the infinitely large are just a few of many possible instantiations of what is, in the end, just the one paradox. Other well know examples include Bertrand Russell's quaint example of Tristram Shandy. Shandy is an obsessive autobiographer. He takes one year to write the detailed account of one day in his life. For mortals with a finite life span, an autobiography such as this can never be completed. However, if Shandy were to live forever, would matters be otherwise? Would there be any day in Shandy's life that is not included in his autobiography?
| My life Day 1 |
My life Day 2 |
My life Day 3 |
My life Day 4 |
My life Day 5 |
My life Day 6 |
... |
| written in Year 1 |
written in Year 2 |
written in Year 3 |
written in Year 4 |
written in Year 5 |
written in Year 6 |
... |
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?
Would there be any day in Shandy's life that is not included in his autobiography?
June 22, September 22, 2021
Copyright, John D. Norton