So-called “infinite sets” do not have any size. They are simply infinite. That is, no matter how many elements you find in the set, there will always be the potential to have found more elements in the set.
That means that there is not a specific number of elements in that sequence and that you can not count or otherwise determine how many elements are in that sequence.
You cannot quantify the number of elements in infinite sets. And as the concept of size quantities how many elements are in a set, such sets have no size. The concept of size does not apply to infinite sets.
Remember what we said about infinity? It does not refer to a number. Nor does it refer to any quantity, it refers to a potential. It refers to the fact that in certain sequences, you must stop at some point.
No matter where you do stop, there is the potential to have progressed further in the sequence.
So, if infinite sets have no size, then there is no basis for comparing their sizes!
And there goes one of the “proofs” that there are different sizes of infinity.
This means that the set of natural numbers, the set of real numbers and so forth, have no sizes. They are simply infinite.
It makes no sense to say that there are “more” real numbers than natural numbers. No, neither set has any size so you cannot speak of one being larger!
There are “proofs” that go something like this:
Take the natural numbers from 1 to 3. There are three of them.
Now consider the real numbers. The real numbers include fractions such as 3.5 and 3.9 and all the numbers between real numbers.
So, if you take all the real numbers from 1 to 3, there are more of them! For instance, there are lots of real numbers between 1 and 2, such as 1.1, 1.5, 1.75 and so forth.
Now take any number of natural numbers from one to X. There are always more real numbers between 1 and X, so therefore there are more real numbers than natural numbers.
This does not work. It does not prove that the infinite set of real numbers is larger than the infinite set of natural numbers.
All proofs that claim to compare the sizes of infinite sets share fundamentally the same flaw: Assuming that infinity is something other than a potentiality and then assuming infinite sequences and series have a size.
But they do not! Therefore, you cannot use them to prove there are different sizes of infinity.
There is no way to get around this. Even if you can identify more terms in one sequence than another. This does not mean that any infinite set is quantifiable or that the concept of size applies to it.
There is no mathematical proof that will ever change this. No math proof can hope to change the nature of infinity or make infinite sets have a quantifiable number of elements.
Infinite sequences do not have a size. Something is infinite, means, it has a certain potentiality. It has that potential or it does not. There is no way to quantify such a concept. It is a binary “It has this potential, this quality or it does not”.
There, since you cannot quantify such a potential, there are no “sizes of potentialities” as we are using the term potential, therefore there are no sizes of infinity.
Talking about “countable” or “non-countable” infinities does not salvage the claim of sizes of infinity. By countability, they are referring to the mapping between elements of sets, not size or quantity.
Mathematicians do not use “infinity” in the same sense used here, which is, a potentiality to progress in a sequence. Their definition of “infinity” is vague and sometimes incoherent.
It is treated as though it was analogous to quantity when it has little to do with the concept of quantity.
What they mean is that some mathematical sequences subsume all identifiable terms within that sequence.
It is not that there is some magical “number” called infinity that comes after such terms. It is simply that all possible terms are a part of that sequence and that no matter where you stop in that sequence, there are more terms after that. That is, always the potential to have continued.
That is the confusion mathematicians have, not seeing it as merely a potential and an indicator that certain sequences never run out of terms.
This leads to a lot of undue significance and mystery being attached to the concept of infinity. When it is a perfectly simple and not a particularly interesting concept.
That is all we have to say about infinity for now. If you would like clarification on this topic, we will happily return to it so that we can provide such.
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