Do Infinities Produce Contradictions?

I love traveling and am approaching two “bucket list” items: visiting all 50 states and all 7 continents. But hands down, one of my most interesting bucket list destinations is Hilbert’s Hotel. Unfortunately, this hotel doesn’t exist, but many of you will recognize that Hilbert used his “hotel” to demonstrate the connection to the infinite. Some philosophers use Hilbert’s Hotel to argue that actual infinities cannot exist in the physical world because basic arithmetic operations involving infinities lead to “absurdities.” However, I think the story is more complicated and I’ll argue that we can’t reject actual infinities based on this understanding of Hilbert’s Hotel. 

Hilbert’s Hotel and “Absurdities”
The German mathematician David Hilbert developed a thought experiment to articulate various aspects of infinite sets. His hotel has an unlimited number of rooms labeled 1, 2, 3, and so on, and every room is occupied. Then, different groups of guests arrive and request rooms. The first counterintuitive aspect of infinities arises when you realize that the hotel can accommodate any number of guests, even though every room is full. If a group needing 3 rooms arrives, the owner simply instructs everyone in the hotel to move to the room with a number three larger than their current room. Everyone can move, and this leaves rooms 1, 2, and 3 open to accommodate the group needing three rooms. Even if an infinite group arrives, the owner instructs everyone to move to the room twice their current room number so that the infinite number of odd rooms are now unoccupied. Though counterintuitive, it makes sense since the number of rooms has no upper limit. 

If we were to put this experiment in equation form, we could write these equations describing the addition of infinite quantities (where ab, and c are finite and  represents an infinite quantity):

a + b = c
∞ + a = ∞
∞ + ∞ = ∞

However, some “absurdities” arise when groups start leaving. If a finite group of 5 checks out, the hotel still has an infinite number of rooms filled. But consider what happens when two different infinite groups leave the hotel. Having the infinite group in all rooms greater than 5 check out leaves only 5 rooms filled. Alternatively, when the infinite group of all even numbers checks out, the hotel has an infinite number of odd rooms filled. In equation form (paralleling the addition equations above), this gives:

a – b = cJust like addition above
∞ – a = ∞OK so far
∞ – ∞ = 5First infinite group leaving
∞ – ∞ = ∞Second infinite group leaving

The last two equations are contradictory! You can’t subtract one value from another value and get two different results. Thus, many people have used this contradiction to argue that actual infinities cannot exist in the physical world. However, mathematicians recognized this dilemma and solved the issue by noting that subtraction is not a well-defined operation for infinites. Lest this strike you as defining the problem away, you encounter a similar solution for a much more familiar mathematical idea. Let me illustrate with an interesting proof: 

A More Familiar Absurdity
For every a (that is not equal to zero), there exists a b (not equal to zero) such that:

a= b
a2= abMultiply both sides of the equation by the same value 
a2 – b2= ab – b2Subtract the same value from both sides of the equation
(a–b)(a+b)b(a–b)Factor each side of the equation
a + b= bReduce equation to simplest form
b + b= bSince a = b, replace a with b
2b= bSimplify
2= 1Since b does not equal zero, we can divide it out 

Clearly, the notion that 2 equals 1 is contradictory so something must be wrong. But what is wrong? It turns out that something we all take for granted causes the problem—namely the mathematical concept of zero. In the “reduce equation to simplest form” step, we are dividing each side by the quantity (a – b). However, since a = b this step entails dividing by zero. Everyone who went through elementary school knows that you can’t divide by zero because mathematicians demonstrated that division is not a well-defined operation for zero. Notice that multiplication by zero works properly, but division by zero results in an undefined quantity. Stated another way in simple mathematical terms, division by zero is not single-valued because when you divide by zero, you can get any number. It is precisely this problem that leads to the contradiction that 2 = 1 and why division by zero is left undefined (or not an acceptable operation). 

Comparing “Zero” and “Infinity”
Notice these parallels between zero and infinity. 

1. For zero, multiplying it by any number (including zero) leaves it unchanged. For infinity, adding any value (even infinity) leaves it unchanged.  

2. For zero, multiplication is well-defined but not division. For infinity, addition is well-defined but not subtraction. 

3. Dividing by zero leads to absurdities. Subtracting infinities leads to absurdities. 

No one I know would say that basic arithmetic operations with zero lead to absurdities. Therefore, actual zeros can’t exist in the physical world. Yet, many people use that same basic argument to claim that actual infinities can’t exist in the real world. 

On a personal note, when I studied infinities for a class in high school (yes, I’m that odd) and came across the bizarre features of Hilbert’s Hotel, my first thought was amazement at how differently infinite quantities behaved compared to finite quantities. Later reflections drew me into a deeper understanding of how an infinite God interacts with his amazing creation. One question that still fills my thoughts is whether God could have created a spatially infinite universe. If actual infinites can’t exist, then the answer is “no.” However, none of the arguments against actual infinites have yet to persuade me that they can’t exist. I realize that Hilbert’s Hotel and its consequences are not the only reason that people claim actual infinities don’t exist and will try to address those concerns in other articles. However, unless we’re willing to reject the physical existence of “actual zeros,” we can’t use the absurdities of Hilbert’s Hotel to claim that “actual infinities” don’t exist.