If a number is infinite (like “Aleph-Null”) then how can there be numbers larger or smaller than it? Shouldn’t that be impossible?

Question

If a number is infinite (like “Aleph-Null”) then how can there be numbers larger or smaller than it? Shouldn’t that be impossible?

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Answer 1

How can there not be a number smaller than infinity? Literally, all of the real numbers are smaller than infinity.

Aleph-null is the smallest infinity, generally called “countable infinity”

It’s the number of natural numbers that exist (1, 2, 3, 4….)

It’s also the number of whole numbers (0, 1, 2, 3, …)

And the number of integers (…-2, -1, 0, 1, 2…)

And the number of rational numbers (1, 3/2, 1/16, etc)

All of these are for math reasons I won’t go into, but we have proven that all of them are the same total.

The number of real numbers (π, e, 1, sqrt(2), 7/8, etc) is also infinite, but we can prove there are more of those than there are natural numbers, an infinity greater than aleph-null. An uncountable infinity.

There’s a lot to go into with the different types of infinity, but the main difference you should know is countable vs uncountable infinity.