Does infinity exist in reality? There are, surprisingly, scientists who think infinity is a possibility even though they are unable to point to any example of infinity in reality. The great mathematician David Hilbert claimed that “the infinite is nowhere to be found in reality.” Nevertheless, the mathematical theory of infinity developed by Georg Cantor is beautiful. Hilbert was in awe of Cantor’s beautiful theory and said “No one shall drive us from the paradise which Cantor has created for us.”
An assumption of the infinite leads to weird counterintuitive results. In this and the following four articles, various ludicrous properties of the infinite are explored. We’ll see, for example, that the entire Library of Congress is encoded somewhere in almost every real number. We’ll also show that the number of points on a line is the the same as the number of points in a square. Does this strike you as weird? It should. And, incredibly, some infinities are bigger than others. By this we do not mean that infinity plus one is bigger than infinity but rather that the number of points on a line correspond to a bigger infinity than the number of counting numbers. And there is no biggest infinity. Given any infinite set, a larger infinite set can be constructed. This is mind-blowing stuff. Cantor thought his math about infinities was so profound he sought audience with Pope Leo XIII to discuss its theological implications.
The Basic Idea
Cantor’s theory of the infinite can be explained, starting with the lowly shepherd tending sheep. Imagine a shepherd who does not count well. He gathers stones until the number of stones is equal to the number of sheep he is tending. The set of stones is said to have the same size, or cardinality, as the set of sheep. If there are ten sheep, the size of the set of sheep is ten and the shepherd picks up ten stones. The size of the set of sheep is the same as the size of the set of stones because there is a one-to-one correspondence. At the end of the day, the shepherd compares the number of stones to the number of sheep. If the number of the stones is the same as the number of sheep, no sheep have been lost. Sheep #1 corresponds to stone 1, sheep #2 to stone 2, sheep #3 to stone 3 all the way up to the tenth sheep.
Let’s apply this same simple idea to the infinite. We’ll show that doing so leads to ludicrous conclusions. Such conclusions are common when dealing with infinities.
Before we begin, let’s clear up a possible misconceptions. The sidewise eight, ∞, is not infinity. As commonly used in math, it means “increasing without bound.” A number can get bigger and bigger and eventually get to one followed by a trillion zeros. Even though this is a really big number, it is not even close to infinity. There is still an infinite number of numbers to go. No matter how high you count, there is still an infinite number of numbers remaining to be counted.
A set with a true infinite size is the set of counting numbers.
The infinite size of this set is said to be the Hebrew letter aleph: ℵ. Let’s play around. Take away the first number in this set, namely 1, to get the set
Even though we’ve made the original set smaller in one sense, both sets have the same size. Think of sheep and stones. Sheep number 1 now maps to stone #2. Sheep number 2 maps to stone #3. Sheep 3 to stone #4 etc. The sets never end so this correspondence goes on forever. There is a one-to-one correspondence between the elements of the two infinite sets so the size of both sets is identical.
This result should strike you as ludicrous. The infinite size of the sets might be the same, but the diminished set is clearly smaller than the set of counting numbers in another sense. This is weird.
This property is the foundation of the saga of Hilbert’s Hotel that reveals how crazy infinity is. If Hilbert’s Hotel has an infinite number of rooms and each room is occupied, there is always room for one more. Move the occupant of room 1 to room 2, the occupant of room 2 to room 3, room 3 to room 4, etc. This frees up room 1 for your new guest. Thus, even though Hilbert’s Hotel is full, there is always room for one more guest.
And it gets even more weird. The size of the set of even numbers is also infinite. The set of even numbers is:
Think again of sheep and stones. Sheep number 1 maps to stone #2. Sheep number 2 maps to stone #4. Sheep 3 to stone #6 etc. This goes on forever. There is a one-to-one correspondence between the elements of the two sets. Since the sets go on forever, the size of both sets is identical.
This has an implication for Hilbert’s Hotel. Even if all the rooms are full, we can still take in an infinite number of new guests. Move the guest in room 1 to room 2. Room 2 to room 4, room 3 to room 6, 4 to 8, and so on. Do this forever. This will free up all of the odd-numbered rooms to now be occupied by the infinite number of new guests. Even though every room is occupied, Hilbert’s hotel can still accommodate an infinite number of new guests.
This, again, should strike you as nonsensical. That’s the problem with infinities. Reasoning with the infinite leads to ludicrous conclusions and is evidence that the infinite does not exist in reality.
Next: Infinity illustrates that the universe has a beginning. The age of the universe is shown to necessarily be finite if ridiculous properties of infinity are to be avoided.
We hope you enjoy this series on the unique, reality-defying nature of mathematical infinities. Here are all five parts — and a bonus:
Part 1: Why infinity does not exist in reality. A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in math. In a series of five posts, I explain the difference between what infinity means — and doesn’t mean — as a concept.
Part 2. Infinity illustrates that the universe has a beginning. The logical consequences of a literally infinite past are absurd, as a simple illustration will show. The absurdities that an infinite past time would create, while not a definitive mathematical proof, are solid evidence that our universe had a beginning.
Part 3. In infinity, lines and squares have an equal number of points Robert J. Marks: We can demonstrate this fact with simple diagram. This counterintuitive result, driven by Cantor’s theory of infinities is strange. Nevertheless, it is a valid property of the infinite.
Part 4. How almost any numbers can encode the Library of Congress. Robert J. Marks: That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math. Math: Almost every number between zero and one, randomly chosen by coin flipping, will at some point contain the binary encoding of the Library of Congress.
Part 5: Some infinities are bigger than others but there’s no biggest one Georg Cantor came up with an ingenious proof that infinities can differ in size even though both remain infinite. In this short five-part series, we show that infinity is a beautiful — and provable — theory in math that can’t exist in reality without ludicrous consequences.
You may also wish to read: Yes, you can manipulate infinity in math. The hyperreals are bigger (and smaller) than your average number — and better! (Jonathan Bartlett)