Curt Jaimungal: What Is Infinity, Actually?
4/7/202617 min
For much of history, many mathematicians—following thinkers like Aristotle—viewed infinity as a never-ending process rather than a completed object. In the late 19th century, Georg Cantor revolutionized this view by treating infinite sets as mathematical objects that could be compared and studied. His work showed that not all infinities are equal, and that there are infinitely many different sizes of infinity. While his ideas are foundational in modern mathematics, some philosophical schools, such as finitism and ultrafinitism, continue to question whether infinite objects meaningfully exist. I subscribe to The Economist for their science and tech coverage. As a TO...
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First 90 secondsCurt Jaimungal· Host0:00
For over 2,000 years, humanity insisted that infinity was only potential. So roughly this means that you can add one more, but you can never actually arrive. Aristotle thought so, Gauss just said so, and then Cantor showed up like a boss with a heresy, and we're still fighting about it to this day. So what is infinity, and why the heck is it so controversial? Cantor treated infinities as objects that are completed in and of themselves that you can grab. Now, not only is that nuts, but he then proved that there's strictly more of these infinities than anyone imagined. Kronecker called Cantor a corrupter of the youth. Poincaré called Cantor's work a disease. Cantor then died in a sanatorium. So what the heck is going on? Now, to understand this, we have to talk about potentiality versus actuality. And no, this isn't Deepak Chopra mixed with set theory. The distinction is sharper than it sounds. A potential infinity is a process, like you keep counting, you do one, two, three, four, et cetera. You don't stop, and at no point do you say that you've had a complete collection. You just say that there's always something else I can do. That's called a potential infinity. Now, an actual infinity says that that whole collection exists right there as a single object. Now, what does that mean that you can handle it as a single object? Well, you can examine its properties.