Skip to Content

Is there a middle to infinity?

This is a fascinating question that has intrigued mathematicians and philosophers for centuries. At first glance, it seems impossible for infinity to have a middle or center point since it represents an unending expanse. However, with some creative mathematical approaches, scholars have proposed intriguing ways to conceptualize a “middle” of infinity.

The different sizes of infinity

To begin tackling this question, we first need to understand that there are different sizes of infinity in mathematics. The smallest version is “countable infinity”, which represents infinite sets that can be counted or put in one-to-one correspondence with the natural numbers (1, 2, 3, etc). For example, the set of all integers is countably infinite.

However, some infinite sets are “larger” than countable infinity. These are called uncountable infinities. The most notable example is the infinity of real numbers between 0 and 1. These can’t be counted or matched up to the natural numbers one-to-one. Georg Cantor proved that this “continuum” infinity is larger than countable infinity.

Cantor’s amazing discovery

In his study of different infinite sizes, Cantor made an amazing discovery: all infinite sets can be compared to see which is larger. His methods allow us to rigorously prove that some infinities are bigger than others. This breakthrough meant infinity had structure and hierarchy, allowing the concept of a “middle” to start making sense.

Does countable infinity have a middle?

For a finite set of numbers like {1, 2, 3, 4, 5}, the middle term is easy to define: it’s the number in the center when they are arranged sequentially. Can we extend this idea to countable infinite sets like the integers?

Let’s try to visualize lining up all the integers sequentially:

-3 -2 -1 0 1 2 3

There is no single integer that could be called the “middle” value of this layout. Instead, the middle is an infinite progression of numbers getting closer and closer to 0 as we approach the center. But never landing exactly on 0.

So in this sense, countable infinity has a “center” but no definitive middle member. The progression of numbers converges towards 0 but does not reach it.

A thought experiment

Here is an interesting thought experiment from mathematician James Propp:

Imagine all the integers written on pieces of paper and placed in a giant line. We remove the 0 paper. Then we insert our hand at the gap and move it across the line, pushing numbers apart as we go. After an infinite amount of time, we would have “centered” the integers on our hand. So in this scenario, our hand is literally the middle point of infinity!

The middle of continuous infinity

Now what about larger infinities like the real numbers? Does the “continuum” have a well-defined middle point?

Going back to the numbers between 0 and 1, there is no single real number that could be considered the center. The value 0.5 has no special status as the middle. In fact, every real number has an equal claim to being the midpoint since there are just as many numbers bigger and smaller than it.

Just as with countable infinity, the “middle” of the continuum is more of a progression, with all numbers approaching 0.5 but never quite reaching it.

The Banach-Tarski Paradox

The Banach-Tarski paradox highlights how describing a middle point of continuous infinity runs into trouble. This famous theorem states that you can take a solid ball, split it into a finite number of pieces, rearrange them, and assemble two balls that are exactly the same size as the original!

This mind-bending result shows that our intuitions about assigning a fixed “center point” break down when dealing with the strange properties of infinite sets.

Infinity has relative midpoints

Based on these analyses, we can conclude that infinity does not have any well-defined absolute midpoint. However, within different models of infinity, there are “relative” centers that serve as midpoints in a limited context.

For countable infinity, the progression of integers centered towards 0 provides a conceptual middle point. And in the continuum of real numbers, any value like 0.5 takes on the role of a midpoint relative to other points close to it on the number line.

So while infinity has no overarching center, it does have perceptible midpoints within the different sizes of infinity. Mathematicians can assign meaningful centers and even calculate distances between these middle points and other values in their given context. This provides a fascinating glimpse into the nuances of infinity!

Conclusion

Infinity’s endless, unbounded nature means that it has no absolute center or midpoint that could be identified. However, different approaches within set theory and topology allow us to study notions of “relative” midpoints that serve as conceptual centers within various models of infinity. By rigorously analyzing countable versus continuous infinities, mathematicians reveal a multifaceted perspective where context determines if and where a middle point can exist.

While infinity has no edges, it does have discernible interior structure. And exploring this structure opens up realms of deep mathematical insight and wonder.