Actually, you repeated yourself 10 =
(0.999…+0.999…+0.999…+0.999…+0.999…+0.999…+0.999…+0.999…+0.999…+0.999…)x0.9999… times to be exact!
.333… = 3/10 + 3/100 + 3/1000 + …
10 (.333…) = 3 + 3/10 + 3/100 + 3/1000 + … = 3 + (3/10 + 3/100 + 3/1000…)
Is that not true?
which can be written as:
3.333…= 3 + 3/10 + 3/100 + 3/1000 + … = 3 + (3/10 + 3/100 + 3/1000…)
So the question is :
Is the fractional bit in 3.333… exactly the same as it is in 0.333… ? Or is it different? Shorter by one digit? 
That is true.
That is deceptive, but not false. The standard means for representation is lacking, thus leading to a deception.
That is exactly the issue.
What you have is:
[10.000… :0R] * [0.333… :3R] = [3.333… :30R]
Those are all of the same cardinality/size. But realize that you have stated that you multiplied EVERY element of the first series by 10. And that means EVERY element, no matter how infinite the list is. To maintain the same cardinality, you cannot have any added elements. And that means that the list of partial multiples will display that EVERY element is 10 times larger. And that means at the end of EVERY member of the infinite partial multiples list, you must have a place saver “0” that came from the multiple of 10 (just like you do with ALL arithmetics).
0.3 → 3.0
0.03 → 0.30
0.003 → 0.030
.
.
.
[0.000… :3R] → [0.000… :30R]
EVERY element must have a zero place holder at the right end, else you did not multiply every element by 10.
That should be pretty obvious.
And that was the FAULT in your reasoning and the shortcoming of the standard method of representing any infinite series of digits.
This is using hyperreals, which it seems you must to make your case. So, fair enough, if we use non-standard mathematical systems, we get a different result.
You had an infinite set and then multiplied it times 10. What did you expect?
And it doesn’t have to be hyperreals.
You had an infinite set of numbers then increased each by a factor of 10. So you still have an the infinite set of numbers. They are merely a little larger. But being larger, you must list them that way and thus maintain that “30” as the last number in the series. And of course, if you do that, when you add them all back together, you will still have that 30 at the extreme of the infinite set.
This sentence does have to be the hyperreals:
There is no ‘right end of infinity’ in the standard reals. c.f. the hotel paradox, showing that infinity plus one is just infinity.
If EVERY member of an infinite set has a “right end”, then the most extreme member also has a “right end”.
You had an infinite set. EVERY member of that set had to have a “0” as its right digit because you multiplied every member by 10.
You don’t throw away place holders when you add them all back together.
And if you count only the right side of the decimal, you still have the same quantity of digits as you started with … as long as you count that last “30” as two digits, else you have a lesser infinity than you started with (which would constitute another Fault).
Fault.
There’s no difference between 1.0 and 1.00. As I argued earlier, there’s an implied infinite string of zeros at the end of every real number that terminates.
If that were true then that would confirm that 1.0 = .999… since the difference between them is 0.000… with ‘no end to be obtained’. In order for the two numbers to be different, you’d have to propose that there is a ‘1’ somewhere out there at the end of all those zeros, and you just said that in fact there is not.
It’s funny how sometimes you can’t get to the ‘end’ of an infinite string of digits and at other times you can say what the last digit is.
I guess it just depends on what better suits James’ argument at the moment.
If the symbol 0.333… means 0.333…333 and also 0.333…330 (and 0.333…300, etc ) then you got a serious problem brother.
There are no lines where the slope =0.Therefore all lines are circular,based on the relative slope of two parallel lines. There are no perfectly parallel straight lines.
Therefore all lines are curved as they approach countless spatial progressions.
Therefore the two curved parallel lines form two similar spheres.
Their similarity is implicit in the law of identity.
Therefore the principle of identity is dependent on the concept of infinite extension. The circle/sphere becomes the ideal representation for the infinite extension of the identity of infinitesimal parts as they approach either 0 or infinity. Here 1=1 is a conceivable and necessay description of tying up the infinitessimal with the infinife. It is a function of the symbolic notation.
Utter nonsense.
When a line is placed in a damping field, when zoomed out in sufficient scale in regards to the resolution, it has a slope of zero (according to the bounds of the resolution.)
a line in "digital code’ can have an absolute slope of zero, as code cannot be zoomed in any further than what it is without ceasing to be code, zero is zero.
An lets say we are not talking about code but space.
Again this utter bonk
A line in a damping field can have it’s sine wave compressed within typical bounds, if the damping field is infinite the slope will never curve onto itself.
And with no damping field there is no guarantee the line will even curve into itself and form a sphere, it will probably form a spiral, and eventually, a randomly shaped blob…
Only a loxodrome caused by a transversal cutting a sphere at the same angle result in appearent straight lines, but that misses the point.
The point is, that nomenclature is to support the ideal shape, - which basically is an approximation to the level of descernibility of near straight pieces which make up a sphere, with the idea, that close to infinite discernibility such differentiations are apprehandable as straight. However, only just before the presumed point of infinity, can such straightness be conceivable.
The presumption becomes necessary, to assume an infinitely large sphere, with infinite radii. This becomes necessary to uphold the idea of an infinite universe.
otherwise infinity becomes just another nomenclature, without ground.
10 times 1.0 is NOT 1.00
You are multiplying EACH member of the infinite set of elements. There are no decimals involved. When you add them all back together to form a single number again, you must maintain the exact same cardinality/size for the set or “count of decimal digits”. And that requires that there be a “0” resulting from your multiplication that wasn’t there before. If the 0 isn’t there, then you didn’t really multiply the entire number by 10.
Quite the opposite.
The difference is NOT expressed by:
1.0 - 0.99 = 0.00
1.0 - 0.999 = 0.000
1.0 - 0.9999 = 0.0000
.
.
.
Thus leading to an endless list of 0s.
The difference between them is expressed exactly as:
1.0 - 0.9 = 0.1
1.0 - 0.99 = 0.01
1.0 - 0.999 = 0.001
1.0 - 0.9999 = 0.0001
.
.
.
Note that EVERY term has a “1” at the end of the string of 0s regardless of how many 0s there are.
Let’s take this one step at a time.
To be correct, there can be no logical contradiction.
A) Agree
B) Disagree
or Forfeit
1.1 times 10 equals 11, there’s no extra zero. Similarly, multiplying every member of the infinite series by 10 gives 3/1, a term that wasn’t there before. But the rest of the series is exactly the same series as before: the sum from n=1 to infinity of 3/(10^n). To say that that that series is actually two different series is problematic.
Countably-infinite and countably-infinite-minus-one are the same cardinality: each element can be uniquely mapped between two such sets. Again, this is the Hotel Paradox.
b
I think for those of us unschooled or unsophisticated in mathematics what makes this exchange fascinating is the fact that with regard to mathematics one would think that there is a way in which to frame an argument such that “all rational men and women are obligated to embrace it.”
But apparently even here that is not really the case at all.
And that is still before the part where those who think they are sophisticated enough to provide the “objective truth” here, are able to reconfigure the “right answer” into a frame of mind that also includes a reference to the world that we interact in from day to day.
Or is that just me?
It all seems somewhat analogous to the mysteries that revolve around the world of the very, very small somehow reconfiguring into the world of the very, very large.
We don’t know exactly how they are intertwined but we do know for certain that they are.
Aren’t we?