For me part of the problem is that dividing by zero itself creates a problem - it is undefined. Dividing by anything greater than zero could not possibly give infinity - infinity is undefined.
I would say that it is grossly or vulgarly defined, not completely undefined.
InfA and H are sufficiently defined to allow normal math operators to make sense. But “standard math” holds to the word and concept of “infinite” to mean merely “endless”, without concern for what degree of endlessness is involved. That distinction didn’t gain authority in mathematics until around 1947 with Hewitt and still isn’t being taught much to many even today. And because of that, many simple minded paradoxes are accepted as nonintuitive truths which are in fact merely poor semantic conflations (e.g. “They are both infinite therefore they are equal length”).
… and welcome back. 
Haha, yeah, I’ve forgotten how we got down this rabbit hole as well. But I think we got to the same place during our last back-and-forth in this topic, so it seems like it matters. And I find it interesting, and trying to convince you has improved my understanding, as it always does.
So I don’t really care why it’s relevant, we’re here and you’re wrong and I intend to fix that.
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This is a pretty unclear definition. As I asked before, can you distinguish the concept of the “range of x” from the concept of the domain of (f(x))? Why couldn’t you provide it? What does it tell you about the elements of P that you don’t otherwise know? Do you have some theorem about how it relates to… anything? What’s the use of this concept?
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I think I see how you’re getting confused.
- You have invented a concept of the “range” of a variable in a function (and it’s distinct from both the common meaning of a “range” of a function and the domain of a function; I’m going to use (range_J) to distinguish it from the common meaning of “range” in set theory)).
- (range_J) is something like the largest and smallest numbers that get plugged into (f(x)).
- Since in this case x has no upper bound, you say the (range_J) is (1 \to \infty ).
- (here’s where the confusion comes in) Since you’ve decided that ( \infty) is the ‘largest number that gets plugged into x’, you assume (\infty) is in the domain of (f(x)).
- You note, correctly, that the function is undefined for (\infty).
- You conclude that the function can’t be a bijection because it’s undefined for certain important values.
The mistake in here is that, whatever the (range_J) is, however important it may be (big clue here to its lack of importance is that it’s being invented here as opposed to in a academic article or similar serious forum), it isn’t the domain, and P having no upper bound does not entail that (\infty) is in the set P nor that (f(x)) needs to be defined for (\infty). It isn’t, and that doesn’t affect whether or not (f(x)) is a bijection.
Still waiting. Keep in mind that (\infty) isn’t an element of P.
You never provided a defined bijection function and I explained why you couldn’t. So asking me to point to something very specific that it didn’t do is nonsense. What does graphiticality not refer to? Your terms were undefined and undefinable in your math model. Your function was nonsense due to the lack of specificity concerning the word “infinite”.
I have explained this many times now. I said before that you never stop bantering, regardless of how wrong you have been shown to be. But for me, you have been sufficiently shown the flaw in your thinking whether you admit it or not.
So unless you come up with a new confusion, this has been enough.
From every element of P. To every element of N. Put up or shut up, James.
I put up many times. That is an undefined function. You can now shut up or step up.
You’ve shown that it’s undefined for something that isn’t in P and isn’t even a number.
So you propose to map an infinite set with a finite variable?
Your function is undefined because you refuse to define its terms, specifically the range of x, although you let it slip that x is “unbound”, aka “infinite”. But if x is infinite, “2x” and “2x+1” are undefined in your antiquated math model.
P has TWO infinite sets. You must have at very least one infinite variable to even try to map it. But when you do that, your function becomes incomprehensible, aka “undefined”.
P is an infinite set, but every element of it is finite. So (f(x)) does not and need not map (\infty).
What you just said is that “(f(x)) does not and need not map endless”. Your sentence doesn’t even make sense.
Fine by me, you’re the one suggesting that it does need to map “endless”.
So (f(x)) being undefined at “endless” is also nonsense. So you objection to (f(x)) is nonsense, and (f(x)) is a bijection from P to N.
Anything for a word game, huh.
What I said was that your x must be endless. And you agreed by saying that it is “unbound”, aka “infinite”. Thus if x is infinite, the term “2x+1” is undefined. And that means that your function is undefined.
Carleas, your recent argument has been that your x has no upper bound and thus no matter what value is plugged into it, there is always and forever another value that is higher, “x+1”. And that is what infinite means - that there is no upper bound, no end. The only problem comes in when you add to or multiply that infinite variable.
Yet when it comes to the idea that 1 = 0.999…, somehow the infinite string of 9’s gets treated as if an end was eventually reached, that it finally got to “infinity”.
My argument all along has been that the infinite string of 9’s doesn’t have an end, upper bound, or terminal destination, and thus cannot ever reach 1.0. It cannot get to infinity. Your argument favoring the use of your x in your function, is exactly the argument that I have been stating concerning the infinite string of 9’s. - There is no end to reach, thus the difference between the string of 9’s and the value 1.0 can never become 0. They can never be equal.
This is just a misunderstanding of how a variable works. Try to take what you’re arguing here and apply it to any other domain where we use functions. Are you really going to argue that every function on the real numbers is undefined?
There’s a distinction between the properties of a set and the properties of its elements. P is infinite, each element is finite. (x) is a variable that stands in for an element of P, not the set itself. For all (x \in P), (x) is finite.
I am not buying that you are that stupid nor that you think that I am, so let’s move on…
By the argument that you just gave and have given several times now, the function that yields 0.999… also maintains a finite variable that never reaches its infinite limit, because “infinite” mean that there is no limit to be reached.
And since that is so, the string of 9s never, ever achieves its limit and therefore can never equal 1.
There are no variables involved with 0.999… unless you use a series expansion to represent it and that’s not necessary.
1/3 + 1/3 +1/3 =1
0.333… + 0.333… + 0.333… =1
All fixed values!
No.
Acknowledge that for all (x \in P), (x) is finite.
P is 2 times infinite plus 1 long.
(P = {N, A})
(N = {1,2,3…} – ) infinite
(A = {a,1,2,3…} – ) infinite + 1
P could have been;
(P = {N_1, N_1, N_3,…})
Then P would be infinite² long.
And S could be;
(S_0 = {P_1,P_2,P_3…} – ) infinite³ long
(S_1 = {S_{01},S_{02},S_{03}…} )
(S_2 = {S_{11},S_{12},S_{13}…} )
(S_3 = {S_{21},S_{22},S_{23}…} )
.
.
And then
(T_0 = {S_1, S_2, S_3…} – ) infinite(^{infinite}) long
And each and every x could still be a finite number.
The problem isn’t x.
The problem is the range of x being doubled, squared, cubed, or added to in any way.
Your “2x+1” term is undefined because the RANGE of x is infinite and you don’t have a “2 * infinite + 1” in your math model. So you cannot fulfill the range of x before the term becomes meaningless. And that is why it fails to map P.
What would your x term be for set (T_0) ?
I know, I know … you don’t know what “range” means, right? Or is it that you don’t know what "2 * " means?
0.333… + 0.333… + 0.333… = 0.999…
As stated before,
1/3 leads to 0.333…
It does not equate to 0.333…
Because 0.333… is not a value or quantity.
This is a different definition of P. If this is the definition, then P is a set with two elements, so it’s cardinality is 2.
I assume this is a mistake, that you really mean that P is the union of N and A. But you’re playing fast and loose with your set construction, and it’s important because it is possible to create an infinity of a different cardinality (this is one of our few points of agreement, we just disagree on when that happens).
I know (range_J) is a concept you made up, from which nothing follows.