No Infinity

Infinity is not quite an endlessly useful concept, but it is a very useful one.  That doesn’t mean that there are actual infinities.  Tom Hartsfield has some mathematical reasons for rejecting infinity as something that can describe the real world.  I am very much inclined to agree.  My favorite example is not mathematical but logical. 

Suppose that there is a grand hotel with an infinite number of rooms (presumably serviced by an infinite number of modestly paid workers).  The hotel is completely booked on Labor Day weekend, when another guest shows up and requests a room.  No problem!  The hotel manager has every guest move to the next highest numbered room, so the guy whose wife kicked him out of the house moves from room one to room two and the meth addict in two moves to three, etc. ad infinitum.  Given the infinite number of rooms, each guest will always be able to move up one.  So the hotel manager can accommodate the new guest.  Likewise, the manager could have each guest move to the next highest odd numbered room.  That would free up an infinite series of even numbered rooms.  Let’s hope he has plenty of towels. 

So: here we have a hotel that is completely booked and yet has room for an infinite number of guests.  That is a logical contradiction and so no such hotel can actually exist. 

I am pretty sure that that is right.  Infinity has intrigued and troubled philosophers since before the time of Socrates.  The most famous example are the marvelous paradoxes of Zeno.  Zeno was a monist who wanted to show that actual change (in this case, motion) involved logical contradictions.  Thus reality was one and unchangeable. 

The simplest of his paradoxes goes like this: Achilles wants to run across the stadium.  That looks easy enough.  Half way across, however, it occurs to him that before he can complete his run he must cover half the remaining distance (one quarter of the stadium field).  After he has done that, he will have cross the next eighth, etc. ad infinitum.  No matter how fast he runs, he will always have some remaining portion of distance to cover.  He collapses in existential despair. 

In fact, Achilles has every reason to believe he can make it.  If he runs at a steady pace, he is covering each portion at half the time.  So if it takes him one minute to cover the first half, what will be true at two minutes?  He will be over the finish line.  One half plus one quarter plus one eighth, etc., is how much shorter than one?  Shorter by an infinite amount, one might say.  Whatever theoretical significance that might have, it has no practical significance at all.  Shorter by an infinite degree is just as long. 

Infinity in either direction, small or large, isn’t real.