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martyb [soylentnews.org] writes:

From the "To infinity and beyond" dept.

Let's talk about big numbers... really, Really, REALLY big numbers. Like infinity. That's pretty big, right? Well, there's infinity and then there's Large Countable Ordinals (Part 1) [wordpress.com] where John Baez explains the concept of infinity and that there are different sizes of infinity!:

I love the infinite.

It may not exist in the physical world, but we can set up rules to think about it in consistent ways, and then it’s a helpful concept. The reason is that infinity is often easier to think about than very large finite numbers.

Finding rules to work with the infinite is one of the great triumphs of mathematics. Cantor’s realization that there are different sizes of infinity is truly wondrous—and by now, it’s part of the everyday bread and butter of mathematics.

Trying to create a notation for these different infinities is very challenging. It’s not a fair challenge, because there are more infinities than expressions we can write down in any given alphabet! But if we seek a notation for countable ordinals, the challenge becomes more fair.

It’s still incredibly frustrating. No matter what notation we use it fizzles out too soon… making us wish we’d invented a more general notation. But this process of ‘fizzling out’ is fascinating to me. There’s something profound about it. So, I would like to tell you about this.

No advanced mathematics are required, but I'll here note that the exposition is pretty mind-blowing. In the "x", "x^(x)", and "x^(x^(x)))" kind of way. And that's barely getting started!

Anyone who enjoyed reading Flatland: A Romance of Many Dimensions [gutenberg.org] by Edwin Abbott Abbott (aka A Square) will find this right up their alley.

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