Not All Infinities Are Created Equal

A mathematician once said that there are more real numbers between 0 and 1 than there are counting numbers in the entire universe of integers. 

Both sets are infinite, both go on forever, yet somehow one infinity dwarfs the other. 

This sounds like the kind of academic nonsense designed to make people feel stupid, until you realize it reveals something profound about how we misunderstand one of the most basic concepts in mathematics.

The set of all numbers is infinite. 

The set of even numbers is also infinite. 

But the set of even numbers is smaller than the set of all numbers. 

This seems impossible because we treat infinity as if it's a number, when in fact it's something else entirely. 

We treat infinity as if it's a number, when in fact it's an idea, a concept that describes something without limits rather than a quantity you can measure or compare in the usual sense.

The problem starts with how we think about "infinity" in everyday language. 

We use it as if it's the biggest number possible, like it sits at the end of the number line waiting for us to count our way up to it. 

But infinity isn't a destination, it's a description of a process that never stops. 

It's not a place you arrive at; it's the absence of arrival.

When mathematicians say the set of even numbers is "smaller" than the set of all numbers, they're not comparing quantities in the way we normally understand comparison. They're comparing the relationships between sets that both continue forever. 

This reveals how inadequate our intuitions are when dealing with concepts that go beyond ordinary experience. 

We're evolved to think about finite quantities, how many apples in a basket, how many days until winter. 

Our brains simply aren't equipped to handle the idea of collections that have no end, so we try to force infinity into the same mental categories we use for regular numbers.

The result is that we end up with paradoxes that seem like mathematical magic tricks rather than genuine insights into the nature of abstract concepts. 

We ask questions like "What's infinity plus one?" as if infinity were just a really, really big number that we could add to. 

But that's like asking "What's blue plus Tuesday?"—it's a category error disguised as a mathematical problem.

The strange hierarchy of infinities, how some infinite sets being larger than others, makes sense only when we stop thinking of infinity as a number and start thinking of it as a property, an idea. 

Maybe the question isn't why infinity seems so strange, that we can think about infinity at all.