…merely your senseless distraction attempts. You seem to be just babbling now.
No one asked me to provide a non-trivial function that operates on the natural numbers … that is the only reason.
And no. I have “ruled out” nothing but undefined infinity references, primarily “2x+1 wherein x is potentially infinite”.
In one post, there are no infinite numbers, then next post, there are potentially infinite numbers and then next post …
Again, each number can never be infinite. The range must be infinite in order to satisfy the bijection. But when that range is infinite, the range for “2x+1” is undefined.
This is fascinating. You aren’t claiming that sets that are elements are the same as subsets anymore, but you won’t acknowledge the distinction either.
I agree this line of discussion is a bit off topic (though it came up in trying to understand the sets you were trying to construct), but it’s troubling that you can’t acknowledge even this.
Hey look, a math(_J) theorem!
In regular math, nothing follows from range(_J).
Well okay, let’s pretend for a while that your reasoning makes logical sense.
P is the set of natural numbers plus an “a”, plus another set of natural numbers. And by your analysis, merely the natural numbers alone will match “one-for-one” to all of those numbers (merely because you can come up with a magic confounding formula that makes it seem rational, despite the obvious flaw).
So okay, your x function is going to match, one-for-one, each and every natural number, the letter “a”, and also another set of natural numbers. So what is the total, from 1 to whatever, range of x"?
Forgiving that you have refused to answer my prior questions:
The domain of x is the set P, so x can be any element of P. I don’t know how your invented concept handles ‘a’. I do know that x is never infinity: the domain is infinite, it does not include infinity. Setting aside the definitional problem, and that nothing follows from range(_J), one can express the natural numbers as an interval, [1,(\infty)), where a square bracket is inclusive and the parenthesis is exclusive.
The limits are all infinity. They are all irrelevant here; a bijection is not a limit.
You are still using infinite series here; you have substituted nine of them for the original one.
There is no controversy here: the equation 0.999999…= 1 merely demonstrates that the two “numbers” are different representations of the same thing. The notion of the sum of an infinite series is sound; the problem is that we have difficulty wrapping our minds around the concept that it NEVER ends. We tend to “feel” that the decimal will always be smaller if only to some very minute amount but, of course, that would only be true if the decimal truncated at some far distant point.
You can only speculate that it is P. I ask you for the RANGE because you determine the range of your own variables.
But obviously you are afraid of the word “range”.
So as x goes to infinity in your math(_C), x converges with both 2x and also 2x+1. An interesting trick. Can you show us the convergence function for that?
And that means that all 3 have the range of x, the natural numbers. So after pairing to all of N, the set A is left out, unpaired, or alternatively, the second half of both sets N and A are left unpaired as well as “a”.
Newly approved post:
You are implying that a conceptual infinity is unverifiable, but does not mean it is inconceivable, right? Which is pretty much an equivocal but uncertain proposition inasfar as muting the question as to whether, there IS infinity, or not, UNTIL which time that it can be verified.
Let’s say, the looking glass of the future discloses a technologically advanced age, where the technical requirements are far in excess of anything which could possibly be imagined today, such as .99999 to the 1000th power specified in some kind of ultra tech , time travel machine, requiring the use of material made up of ultra stress potential, to withstand the tremendous forces incurred in time travel.
Here it would be within 1/0.99999999999999999999999999999999999999999999999999 of differential of a functional variable, and possibly with no end in sight. Does this not bring the idea of a continuation of this difference between the sum of the parts to be as split between 1 and and 1-.99999999999999999999999999999999999999999999999999?
Until that future time, is it safe to say that the answer to infinity as an open system is closed?
The answer could be yes, only if the answer is based on a probable set. Is there any legitimacy in any case, one way or the other, to qualify a certainty before that time?
In the above example, .99999 was carried only to n=50 th decimal point, whereas n=1,000 was used in the narrative. How close does this come to what has been described as ‘hyperreal’, depends on the definition of what is considered to be ‘real’.
The same kind of analysis can be used between limitless and limited functions, it is limitless until a limit is used. The difference between the two types of propositions, is the first is a quantifiable and the second is a qualifiable one.
The former uses numerical functional predecessors of at least .1, whereas the later uses the presumption of the sum of all partial differentiations=1 .
If presumption for the second is to hold water, then that set is inconclusive as to whether the sum can ever be assumed to be 1, because an infinity of .9’s has to be assumed. The other kind of infinity only proposes, that in the function of n sequences of 9’s, where n~infinity, there can not be any closing 0 number.
Therefore the proposition that n number of sequences of .9’s where n=1~infinity, is only a linear real set, whereas the set described as a sum of all members of the set, where the same proposition leads to the sum of all the differentiated members of the set are not linear but differential.
The two types of sets are not differentiable by the way their functions are defined. The former function only points to linear succession , of a simple linear sequence of the type- x~x~~…whereas the second involves the type- : the sum of the above =x+x+x…+x=1.
Hypothetically, a linear sequence, proposing nothing else then the ongoing sequential progression of a addition of a number which diminishes value by a set constant rate, does not imply a different(ial) function
that has a limit.
If that different function could be inferred, that would require the function limit the function into a real limit using real numbers. The problem with the difference between real and hyper real numbers are, that they can not be said to differ except , how their function , or use be relegated.
It is like saying, any differential value has a real based set, until that function becomes useless. But since derivatives can only be derived backwards, in terms of differentiating the useful from the useless, until that time to talk of useful functions makes no sens. But that is all that differentiation can do, it can not work forewards. Until that time, all math is merely intuitive.
Yes, we have all read Wiki and heard from the priesthood. We know what we are supposed to believe to be the holy truth from above that our small minds are just too inept to conceive. But interestingly throughout history, in almost every case, the priesthood has turned out to be, if not entirely wrong, at least distorted and in need of correction … which eventually comes after centuries of debates similar to this wherein the outsider is continually accused of being just too inept to see the greater wisdom of the holy enlightened priesthood.
Frankly, I prefer pure logic. And it seems that pure logic dictates that 1 and 0.999… are entirely different things, for several reasons.
True.
Some people do.
That seems to be contradictory. Please believe that nothing that I am arguing has anything to do with “feelings” (other than attitudes towards certain obnoxious debaters).
The infinite series summations dictate that there is ALWAYS a smaller amount to be added (that is what “infinite” means) in the attempt to resolve the ratio of 1/3 into decimal form and the function that leads to 0.999… It has nothing to do with what we poor mentally backward, decrepit souls “feel”.
I would submit that those following their “feelings” rather than their rationality are those who out of hand, without a single rational argument, accept the holy preaching that 1 = 0.999… simply because “THEY say…”. They are the ones thinking in terms of a truncation that isn’t there, because only with a truncation can the series ever reach 1.0.
What’s your position on this today? 
0.999… is not a fixed value? 0.999… is a fixed value but not equal to 1?
0.999… is not a quantity or fixed value. It is an endless series of diminishing decimal values. The ellipsis “…” means “infinitely” - “never ending”. The decimals begin but never end - open ended. The fraction is never satisfied by the series.
1 is an obvious quantity and fixed value with a beginning and an obvious end.
So, sometimes when you divide two fixed values(1/3, 2/3, 1/9, etc) , the result is not a fixed value. It’s “something else”.
Yet oddly enough, the rules of mathematics seem to deal with it adequately. It doesn’t break any theorems. Math doesn’t come to a grinding halt at these particular operations. 
What is so shocking about that?
The distinction between and infinitely diminishing series sum and a fixed quantity is irrelevant to science or any practical concern. The logic of the distinction doesn’t interfere with any normal mathematical operations. The existence of the entire hyperreal system of reasoning changes nothing concerning practical issues. It is entirely an issue of keeping your mind, language, and logic coherent. It is a pedantic concern, not a life threatening concern, much like arguing whether there is life in distance galaxies.
According to you, one can’t even write this :
1/3 = 0.333…
Because it’s not true. It’s not equal. 0.333… isn’t even a number.
Sounds like a major problem.
Well, either 0.999… =1 or it doesn’t (as Iambig might say) and that has some effect on math results. You can’t just wave it off. You can’t wave off a fundamental problem with division (1/3 =/= 0.333…). Where is the rigor then?
Coherent??? 
You just whacked a bunch of coherence in math, if your analysis is correct. You just broke a bunch of math properties.
Being pedantic, you are correct. The exact truth is that 1/3 => 0.333… {leads to, not equates to}
But no one has serious practical reason to care. And an argument from pragmatism is fallacious. Truth doesn’t depend on how practical the conclusion might be for society (except to the priesthood).
The only incoherence being corrected in common math leads to the inclusion of the proven hyperreal system. And the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities. Such can be merely stated in a foot note. Nothing very serious changes anywhere.
The provable incoherences in Relativity Ontology and especially Quantum Physics are far more serious.
It’s unclear what “leads to” means. You need to define it.
Then they are not numbers or quantities in the hyperreal system either.
You have 3 possible results in the hyperreal system:
Pick your poison.
Yeah right. And I suppose that you don’t know what the word “range” means either.
The hyperreals are about degrees of infinite, not quantities.