You may have read this title and went “Huh? Why is this about sizes of *infinity*? Isn’t *infinity* just…*infinity*? It’s the concept of something that goes on and on and never stops? It’s not a number, so how can we talk about size?”

Well, you can thank a gentleman named Georg Cantor for this discussion. He was the founder of set theory, and he showed that the infinite set of real numbers is actually larger than the infinite set of natural numbers. If both go on forever and ever, how does one compare the sets?

Cantor had an ingenious explanation. Take an infinite set such as the whole numbers {0, 1, 2, 3, 4, …} and an infinite set such as the natural numbers {1, 2, 3, 4, …}. Each element in the whole numbers can be paired with an element of the natural numbers.

0 –> 1

1 –> 2

2 –> 3

And so on.

There is a *one-to-one correspondence* between the elements of the 2 sets. Also, each set is *countable*. You can assign a counting number to each element in each set. These 2 sets are the same size.

Now, make the same comparison of the natural numbers with the real numbers.

1 –> 0.12345…

2 –> 1.45286…

3 –> 2.93789…

4 –> 3.19887…

And so forth. Try to match each natural number with each real number. Just when you think you’ve covered them all, take the tenths digit of the first real number, the hundredths digit of the second real number, the thousandths digit of the third real number, and so on to get 0.1578… Now add 1 to each digit to get 0.2689… Uh-oh! You get a new number that isn’t on that “exhaustive” list of real numbers! This new number is different in the tenths place from the first number, different in the hundredths place from the second number, different in the thousandths place from the third number, and so forth. You will never find a real number in the “exhaustive” real number list that is exactly the same, because it will differ from this new number in at least one decimal place. We found a real number that wasn’t included in the list of real numbers! It’s a paradox, and it shows that the real numbers are uncountably infinite. And this infinity is bigger than one that is countable. It can be demonstrated as a continuum on a number line, while a set that is countably infinite cannot.

So, Cantor showed that some infinities are larger than others. From there, he couldn’t decide if there was another size of infinity between the size of the integers and the size of the real numbers. He came up with the Continuum Hypothesis, which says there is not. But we don’t know for sure. It’s at the top of the list of unsolved questions in mathematics to this day.

The fact that we can’t answer this one particular question doesn’t stop mathematicians from asking more related questions and seeing if those can be solved or even lead to the sought-after answer. One such question was recently answered and made the news. Mathematicians were asking whether 2 orders of infinity, **p** and **t**, were equal or not. These 2 orders are the minimum sizes of collections of infinite sets of the natural numbers with particular properties, and both are larger than the set of natural numbers. If someone showed that **p** < **t**, then this would prove the existence of an infinity between that of the natural numbers and that of the real numbers, thus disproving the Continuum Hypothesis. What happened? They proved that **p** = **t**. So, the Continuum Hypothesis still stands and remains unproven. But the work will continue. Some mathematicians feel unsettled by this result. They have a feeling that more infinities exist, but so far they just keep coming up with the same 2. They won’t let it rest until the Continuum Hypothesis is proven or disproven.

If you would like to read more about this recent discovery, check out the article in *Quanta* Magazine at Mathematicians Measure Infinities and Find They’re Equal