A Case Against the Logic of Having an INFINITE Number of Things

[Here’s the way it starts…]

(First spelled out by German mathematician David Hilbert (1862-1943))

Part I: Imagine a hotel with a finite number of rooms.  Suppose all rooms are filled.  As a new guest arrives, the Proprietor (P) says “Sorry, all rooms are full.”  End of Part I (and the story).

[That is, unless you risk changing the “scenery.” Suppose the story goes a different way. But for that, which we’ll call Part II, you’ll need to walk through the DOOR.] 


Part II: Now suppose that, instead, in the hotel there is an infinite number of rooms (#1, #2, #3 . . .) [using “#1, #2, #3 …” is one way to represent infinity since the “3-dot ellipsis” suggests the numbers, in the order given, could go on forever], and all rooms are occupied–not a single vacant room.  Now another guest arrives and asks for a room. “But of course!” says P, and immediately he shifts the person in #1 into Room #2, the person in #2 into #3 and so on out to infinity.  Now Room #1 is vacant and the newcomer is housed, even though before all rooms were occupied!

Suppose, however, an infinity of new guests shows up at the desk, each person asking for a room.  “Of course, of course!” says P, and he shifts the person (p) in Room #1 into Room #2, the p in Room #2 into Room #4 , the p in Rm #3 into Rm#6, etc. out to infinity, always putting each former occupant into a rm with a number twice his own. Because a number multiplied by 2 always equals an even number, all guests end up in even numbered rooms!  As a result all the odd-numbered rooms become vacant, so these can accommodate the infinity of new guests, despite the fact that the hotel was already full!  For the record, P could repeat this process an infinite number of times.

But now suppose that guests in rooms #1, #3, #5 . . . check out.  Now an infinite number of guests have left and half the rooms are empty.  But a half empty hotel looks bad for business.  No matter.  By shifting occupants as before, but in reverse order, P transforms his half-vacant hotel into one jammed to the gills!  Thus you might think that P could always keep this strange hotel fully occupied.  But you would be wrong.

Because then if guests in Rooms #4, #5, #6 . . . check out, at a single stroke the hotel would be virtually emptied, leaving only 3 registered guests, and infinitude would be converted to finitude.  And yet it would remain that the same number of guests checked out this time as when guests in Rooms #1, #3, #5 . . . checked out!

In both cases we subtracted the identical number of guests from the identical number of guests and yet did not arrive at an identical result.

In short, a suitable sign for Hilbert’s Hotel would be “NO VACANCY, GUESTS WELCOME.”  As far as dealing with real, tangible things, Hilbert’s Hotel is absurd.

This illustration is a small, small part of William Lane Craig’s argument from philosophy (in Reasonable Faith: Christian Truth and Apologetics–an eBook for $7.99, that has much, much more, including arguments from up-to-date science) in his case that an infinite number of things cannot exist, which implies a Beginning, and that something or someone outside the physical universe had to start things off.

Note further that the argument presented here against an infinite number of things is also supported by our “Universe Size” that we posted on Mar. 25, 2013.

[For our category “SOURCES,” we’ve said little, because the logical question, “Sources for what?” we haven’t addressed. For now we’ll say, that  the sources we recommend are those that we consider overall to be intelligent, reliable, and useful for understanding matters that relate to Christianity, philosophy, and science. Craig’s book, underlined above is one of the books we strongly recommend for study and reference.]



Author: John Knapp