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?
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>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.
Not in and of itself. The way you’re defining the set may require it to include all real numbers, but you can define the set in other ways that exclude some real numbers. Depending on the exact definition, such a set can still be infinite.
For example, let’s take the set of all real numbers. This is infinite. It has to include all real numbers, but only because that’s the rule I defined. A slightly more complicated example would be all complex numbers: all real numbers are complex numbers, so if I want a set of complex numbers it has to include all real numbers too.
But let’s change up the rule a little bit, and just take the set of all real numbers between 0 and 1. This set is also infinite: in fact, it can be proven that it’s just as large as the set of all real numbers. But it doesn’t include 2, or 1.5, or π, or many others: in fact, almost all of the real numbers are NOT in this set. And despite that, it’s still infinite.
0, 0.1, 0.01, 0.001, 0.0001, etc.
You can keep going on for forever, and you will always be bounded between 0 & 0.1.
0, 0.01, 0.0001, 0.000001, 0.00000001, etc.
The above still goes on forever, but it skips 1 decimal place every time, so it’ll include every number from the first list, but only ~1/2 from the first list will appear in this list, yet both are infinite.
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