Agreed that I should've used (or at least mentioned) cardinality, but that doesn't mean one is bigger than the other.

Think of it this way:

We have two infinite sets: A and B. Let us imagine a process (isn't that word loaded after reading this thread...) wherein B systematically takes its elements and brags about them to A. Once it uses an element, it will never show it again.

If A, in turn, can counter anything B shows with an element of its own that it has never used before, then:

|A| >= |B|

And, if we reverse this process, with A showing its elements to B, and B counteracting, then we have:

|B| >= |A|

Well, if both of those things are true, the only alternative is:

|A| == |B|

THIS IS NOT A PROOF. This is a way of thinking.

The problem people have is this thought:

"If every element in A is also in B, but B has elements beyond that, then B must be bigger."

This is a symptom of finite sets, not an actual rule. We equate them in our heads because there's a 1-to-1 relationship in the finite space, but they are not the same.

This isn't a rounded value; no rounding is happening. It is just a different name. Talking about something else entirely doesn't really justify anything.

Not really. We have two different notions of size that can be extended into the infinite case consistently. Intuition underlies all of our definitions in math, in infinite cases or otherwise.

It still seems that all you're saying here is that the notion of size I'm referring to does not align itself with the notion of size as cardinality. I don't argue that - I'm only saying that there is a sensible and consistent way to state that strict supersets are strictly bigger than their subsets, even if they have the same cardinality. Clearly this isn't a property that's preserved by bijections. What you've put forward is the claim that if there is an injection from one set to the other, and an injection from the other set to the first, then we have a bijection and consequently they have the same size (Schroeder Bernstein theorem, yum). But that's just to equate size with cardinality.

Well yes, but these things get left behind with time. Definitions are motivated by intuitions, but intuitions in time often are stripped away in favor of rigor (speaking of loaded words >.>...). While intuitively one seems larger than the other does not mean that it is larger than the other as again I just don't think intuitive and infinity have much in common.

eh.. you seem to be missing the point. Without intuitions, definitions would be arbitrary. There's a reason why our definition of a metric has the three properties it does - because those are the properties that we associate intuitively with distance. The same thing applies to all definitions - including the definition of the size of a set. In fact, that's why we tend to speak of cardinality rather than size when it comes to infinite sets, yet we normally just talk about the size of a set in finite cases.

What is the limit as x goes to positive infinity of the function (1/x)?

Is the answer 0.000000 ....... 00001 equal to zero?

The answer is exactly 0. 0.00...001 is either a finite decimal expansion with a 1 at the end, in which case it's strictly greater than zero and thus not the limit of that function, OR it's a broken piece of notation that cannot refer to an actual decimal expansion because there's no place to put the 1.

Totally tried explaining that earlier. There's no end to put the "1" at.

Well I don't disagree with you on all of that, I am just saying they are the same size... because they are the same size. Intuition is another one of those loaded words - I don't think a fifth grader would find anything we're talking about as 'intuitive.' Not all concepts in mathematics are intuitive - they require some reasoning out. Your argument makes more sense to me with reasonable instead of intuitive but we're really just arguing semantics here at this point.

the representation is ALL YOU HAVE. decimal IS a method of representing. if i'm saying that's not ideal then yes i'm saying the number itself is a problem.

"green is bad"
"green is just one way of seeing this color"
"no green IS the color"