Originally posted by Adrift
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Dr. Craig's problem, when discussing infinity, is that he constantly makes the mistake of trying to use infinity as a number. Infinity is not a number, and as such, you cannot perform mathematical operations on infinity. There are numbers which are infinite, however, and one can perform mathematical operations on these numbers. Unfortunately for Dr. Craig, these numbers can resolve the alleged problems which he cites for infinity fairly easily.
For example, insofar as Hilbert's Hotel is concerned, let's say that the Hotel in question has N rooms, where N is an infinite Hyperreal such that N=(1,2,3,4,5,...). This defines a situation in which the Hotel has the same number of rooms as there are Natural numbers, and let's say that all of these rooms are occupied by a single patron. That means there are N patrons staying in the Hotel. Now, let's say 3 people check out of the Hotel. Sure enough, the number of patrons remaining in the Hotel is still infinite, but it is not the same number of patrons as it had originally. Now, there are N-3 patrons, where N-3=(-2,-1,0,1,2,...). It is not true that N=N-3.
Now let's say that instead of 3 patrons, all of the patrons in the even numbered rooms check out. Sure enough, this means that an infinite number of patrons are checking out of the Hotel-- but again, it is not the same infinite number as we had originally. In this case, we see that N/2=(0.5,1.0,1.5,2.0,2.5,...) patrons have left, and again, it is not true that N=N/2.
Dr. Craig doesn't have a very good understanding of the mathematics which he attempts to describe.
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