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

Usually because you have a reason to believe they are larger or smaller based on the rules that still apply.

For example, exponential divergent functions (2^x) grow faster than linear divergent functions (2*x). If you chose x to be infinite then whatever undefined thing (2^infinite) would be, it would definitely be bigger than infinite*2 or you break the rules of your basic operants.

You can also take a look at different classes of infinity. For example, there is an infinite amount of integers (1,2,3,4,…)., In theory you could use them to count an infinite amount of objects right?

Now take a look at real numbers (1, – 2.35894, 2/3, pi,…) and try to count them all. 1.0 is the first, 1.1 the second and then.. How to proceed with the third? Is it 1.2? Or 1.11? Or 1.100000001?

It does not matter because even between 1.00000000001 and 1.00000000002 there is still an infinite amount of real numbers. You can just keep padding zeros indefinitely and we haven’t even reached 2 yet. It can’t be done. There are more real numbers than can be counted with an infinite amount of integers.
Clearly there must be more of them by an entire order of definition.