Does infinity necessarily need to include ALL real numbers?

Question

In other words, could you say that there is an infinite set of real numbers that are NOT a particular real number x – hence “infinity” does not necessarily need to contain all real numbers?

In: 3

Answer 1

>Does infinity necessarily need to include ALL real numbers?

Nope.

>In other words, could you say that there is an infinite set of real numbers that are NOT a particular real number x – hence “infinity” does not necessarily need to contain all real numbers?

Yep. The set of integers (whole numbers) is infinite, for example, but it only contains integers and not other real numbers.

We can even say that some infinities are “bigger” than others. Integers are countably infinite, in that you can actually count them, you can go 1, 2, 3, etc. where as real numbers in general you cannot count them because where do you start? We therefore call the set of real numbers uncountably infinite.