Can you go beyond infinity

Cantor's way of comparing the size of sets is the criteria used by most mathematicians. So, how does this relate to your question? Cantor showed that no matter how large a set you have, you can always produce one that is larger. So, what's larger than infinity? Another infinity that is larger than the one you started with. Children often wonder what the biggest number is, usually settling for the biggest whose name they know — a hundred, or a thousand.

Later they realise that whatever number you choose, a gazillion, say, it can always be trumped by a gazillion and one. So we have to come to grips with it, rather than dismissing it as nonsense.

Around AD, the philosophy and mathematics of infinity became entwined with early Christian beliefs. In AD Eunomius argued that creation as a whole is finite. Consider the most perfect possible being. Since a being that exists is more perfect than one that does not, the most perfect possible being must exist. Especially when the property is existence itself. A major reason is a profound discovery about the infinite, one that all the philosophers and mathematicians, for that matter had missed, made by Georg Cantor in He demonstrated, logically and rigorously, that even within the realm of numbers, infinity comes in different sizes.

We learn about the counting numbers 1, 2, 3… as children. Cantor proved that the infinitude of all such decimal numbers is greater than that of the counting numbers. To mathematicians, however, there are only two basic operations: addition and multiplication. We get subtraction and division for free, as the opposites of addition and multiplication. More specifically, every subtraction question can be thought of as an addition question, and every division question can be thought of as a multiplication question.

For instance, a typical multiplication question might look like:. But what does this mean? In other words,. So a division problem is really a multiplication problem with the unknown number on the other side of the equation. But there is no number that works! Since multiplying any number by zero gives you zero, it follows that zero divided by zero could be literally any number at all. Since every rational number can be thought of as a pair of natural numbers, we can draw up the following infinite table to capture every possible rational number:.

We only want each fraction to appear once in the table, and at the moment this table captures each fraction over and over again, infinitely-many times. So we take care to eliminate all the duplicates by crossing out each entry in the table where the fraction can be simplified all the entries where the numerator and denominator share a common factor are removed. Having done that, we can now zigzag our way through the table to catch every single fraction, like so:.

Then all we have to do then is list every fraction in the order we encountered it during the zigzag:. We can check that every natural number has a partner the list of fractions never ends and that every rational number has a partner every rational appears exactly once in the table, after we carefully remove the duplicates.

But Cantor had one final trick up his sleeve, which he published in his famous paper. Unlike cardinal numbers 1, 2, 3 and so on , which tell you how many things are in a set, ordinals are defined by their positions first, second, third, etc. But one of the things about numbers is that you can always add another one at the end, said Towsner. And since counting is kind of like adding additional numbers, these concepts in a way allow you to count past infinity, Towsner said. The strangeness of all this is part of the reason mathematicians insist on rigorously defining their terms, he added.

Unless everything is in order, it's difficult to separate our normal human intuition from what can be proven mathematically. For us mere mortals, these ideas might be tough to fully compute. How exactly do working mathematicians deal with all of this funny business in their day-to-day research?

Originally published on Live Science. Adam Mann is a journalist specializing in astronomy and physics stories. He has a bachelor's degree in astrophysics from UC Berkeley.