If it’s not a value or quantity, then what is it?
Different than what?
And I don’t care what “cardinality” you place on it. We are discussing the bijection between P and N.
And no, I don’t mean the union. You already tried that. P has two infinite subsets.
More BS. We aren’t even talking about cardinalities. And you would have to be very seriously stupid to not know what “range” means, especially since it has been explained several times.
It is an endless string of digits that results from trying to represent a fraction in decimal form. The endless string never reaches a satisfaction point where it would finally form an equivalence to 1/3. It is a misrepresentation of the fraction 1/3.
You’re confusing a map and the terrain. The “endless string” is the map. The map represents a fixed value.
It represents the exact value 1/3 in the real number system.
In a hyperreal number system , at best, it has a value 1/3 minus a fixed infinitesimal.
Different from ({1_A,2_A,3_A,…,a,1_N,2_N,3_N,…}). Your original set had an infinite number if elements. By your more recent definition, it has 2 elements.
A set that is an element of another set is not a subset. A subset is a collection of elements of a set. For example, if (Q = {1,2,3}), and (R = {Q,4}), (2) is not an element of (R), and (Q) is not a subset of (R) (it’s an element).
By contrast, if (W) is the union of two sets, (W = {1,2,3} \cup {4}), then (2) would be an element of (W), and (Q) would be a subset of (W).
These differences matter, James.
It has, poorly. But enough to show that it’s just a concept you pulled out of your ass about which there are no theorems that apply to the current conversation. Nothing follows from (range_J).
BS semantic games … again.
Face it. You are wrong.
There is no bijection between P and N.
Assuming you mean your original definition of P (since I agree there’s no bijection between a set of 2 elements and an infinite set), there is, the one I provided, the one for which you can’t point to an element that isn’t mapped.
Let’s try a different way.
Is it true that, if x is a natural number, 2x is a natural number?
Is it true that, if x is a natural number, 2x+1 is a natural number?
Not if x is infinitely large (using YOUR math model).
Actually, you are demonstrating how wrong your own math model is.
Then there must be a number such that 2x is infinite but x is not infinite. What is that number?
OK, so looks like there’s a preliminary question I foolishly neglected to ask:
Are there infinitely large natural numbers?
EDIT: Sorry, missed Phyllo’s post. I think his question is in a similar vein.
In the simple minded math model, if a number is infinite, there is no definition for 2 times that number, 2 times infinite.
Carleas, of course there are. But wake up. It doesn’t matter if the x is a natural number. What matters is that x is potentially infinite. And worse, you depend upon it being potentially infinite in order to satisfy your bijection function.
Then because x is potentially infinite, 2x+1 is undefined.
What is the limit as x goes to infinity for:
A) x?
B) 2x?
C) 2x+1?
And that is BS. Any and every set has subsets. That doesn’t mean that its size is merely the number of subsets.
James, you can define whatever math you want, but your position is confusing when you redefine commonly used terms for your uncommon meanings. There’s nothing inherently wrong with defining a set that includes both the counting numbers and also some set of infinite numbers, but it seems willfully confusing to call that the “natural numbers”. The natural number, as that term is commonly used, are understood to be closed under multiplication and addition, and don’t include any infinite numbers.
Again, a set that is an element of a set is not necessarily a subset of that set. Those are different things. Similar to the above, you’re confusingly using common words in this weird Jamesian math that doesn’t seem necessary or valuable.
So it’s pretty clear now that we aren’t even talking about the same sets or operations or anything, because you’ve been using common words in uncommon ways and you don’t actually mean to say what a reasonable person with a background in math would understand you to be saying. Awesome. 1 = 0.999… in math, but maybe not in math(_J).
Did you not read my post correctly? I specifically said that X IS NOT INFINITE but 2X IS INFINITE because 2x is larger than x.
What I’m reading in your posts is that there is a range of finite x when 2x is finite and there is a range of finite x when 2x is infinite and a third range when both x and 2x are infinite.
You should be able to identify the values of x at the transitions.
Then you are saying that every function is undefined in the natural, integer, real and complex number systems.
BTW, is f(x)= 1*x also a function which is “undefined”? 
The set of natural numbers is infinite. The SET is infinite. And that means that the natural numbers extend infinitely. And when you assign “x” to represent every natural number, the RANGE of x is infinite. And that makes the range of 2x to be 2 * infinite - undefined.
Again, BS.
Give me an example of a subset that is an element of a set S yet is not a subset of that same set S. Every element in every set is a “subset” of at least size 1 (not that such is relevant in this issue).
No, what is pretty clear is that you wish to lie about me making up definitions so as to escape the fact that you erred.
No single number is infinite. Sets or ranges can be infinite. The numbers aren’t. A number can’t be endless (except for those damn “0.333…” and “0.999…” type “numbers” that aren’t really numbers).
No. As long as the range of x is finite, 2x will also be finite. And when the range of x is infinite, 2x is undefined.
Now Carleas, answer my question:
Happy to: Lets say S is the set of all finite sets. In that case, the set T = {A,triangle,3} is an element of S. However, A, triangle, and 3 aren’t elements of S, because they aren’t sets. So T is an element of S, but it isn’t a subset of S because the elements of T aren’t elements of S.
Okay, so you just said that there are no infinite numbers. Therefore, no problem with the functions as used by Carleas. The fact that a range is “countably infinite” does not mean that it contains any infinite numbers.
Good, we can move on. 
They ARE sets. They are sets with size = 1.
There certainly is a problem with his f(x).
For his f(x) to be a complete bijection function between P and N, the x must represent the entire endless set of natural numbers. That alone would be fine. The variable x being endless isn’t an issue alone, but the fact is that his function contains a term “2x+1”, which must represent twice endless plus one.
“Twice endless plus one” is a meaningless term unless you are in the hyperreals, or at least in logically sound math.
You’re being pedantic, but since I truly believe you can’t come up with an example on your own, I’ll humor you:
Let S be the set of infinite sets, and A is the set of natural numbers. A is a element of S, but not a subset, because e.g. 1 is not a member of S.
I note that you haven’t responded to Phyllo’s point that, were this true, all functions on the natural numbers, integers, real numbers, or complex numbers would be similarly undefined.
Again, it’s math(_J), not math.
I am merely being accurate and you are trying to make excuses.
I responded by clarifying the issue (which you will eternally deny, of course).