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.
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.
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.
You begin with an endless string of single digits (the digits that comprise the endless series of decimals; “0.999…” or “0.111…” or “0.333…”). You have no option but to add a “0” placeholder to each and every one.
Last chance, Carleas:
Please state your case for how a correct conclusion can be be drawn from logical contradiction.
Do you really want to try to go through the rigor to prove that? You will fail.
You multiply an infinite series of digits by multiplying each individual decimal digit and carrying (as with all arithmetic), then seeing where it logically leads as the process is presumed infinite.
I don’t trust that answer. What do you mean by “context of mathematics”? If you are referring to whatever the consensus of mathematicians might currently think, that is not good enough. Logical consistency is independent of what anyone might be currently believing concerning any subject.
Turns out that was easy as fuck, not sure why you are having so much difficulty with it.
I mean, I’m not talking about art or poetry or philosophy of religion or whatever. I’m saying, in the context of mathematical thinking and reasoning, a logical contradiction shows that either a starting point is false, or the reasoning employed is flawed.
I don’t have difficulty with it, but you will shortly.
Okay.
Next Question:
The quantity of digits to the right of the decimal in the number represented by “0.333…” is equal to the quantity of integers (since they are produced one at a time through long division and thus merely counted with integers). And for convenience we can call that a set of quantity “infA”. InfA is the number of elements in that set.
Only through an intuition of math/reason, can a reductive differentiation of two types of logic manifest a connection.
Difference through similarity-familiarity, and identity through reason if, not exclusive, then at some point may share a functional derivative. The difference at both absolute limits, are somehow limited without violating the definition of the limitless. They are at once limited and limitless at this point, and in spite of contradiction of nomenclature such contradiction resolves. Perhaps this is because the way the contradixtion effects and changes, as do quantum effects seem to change by the effected quantum particles.
Because of Carleas objection of not introducing non mathematical considerations, i withdraw my response to James quiery into logical contradixtion, but let stand as a valid conjecture holding that a purely mathematical demonstration will lead into that very contradiction.
Here, we have to be careful. InfA is not a number on the standard real line. It’s true that the set of decimal places of .333… is of the same cardinality as the set of integers. But calling that cardinality a “quantity” or a “number” are being a little too loose with language, especially since it looks like you mean to build a rigorous syllogism. I applaud that effort, but if that’s the goal, we should be wary of accidentally introducing ambiguities.
Does that answer the question, or if not, is there a way to reword the question to address my concerns? Or perhaps my concerns are misplaced?
We have previously agreed that infinites can be of different sizes. One particular size can be ("may validly be ") set as a standard with which to compare others. Obviously if there are different sizes, there are differences between those sizes to be potentially measured.
I am giving one particular infinite size a name, “infA”, to be the size of the set of all integers. InfA does not have to be called a “number” or a “real number” or a “quantity” or anything other than a standard size (aka “infA elements”) of the infinite set of integers.
So we have agreed on the 2nd question of the set of digits to the right of the decimal being of size infA. Or that there are “infA elements in the set”.
Therein lies your fault. You treat something that is not a number as a number. Which makes you think that there is a zero at a particular position on the end. In fact, there is no end.
Yeah, I’m with Phyllo here too. Your reluctance to talk about it as a cardinality rather than a size suggests that you mean something different. Is that right? My concern is that when you call it a ‘size’, there will be a temptation to make an equation like
InfA + 1 > InfA
which is not the case if InfA is the cardinality of the integers (and 1 is the cardinality of a set with 1 element).
And again, I mean to be pedantic here, because I expect the syllogism you are building to be sensitive to the details. Cardinality is a specific mathematical concept; ‘size’ is ambiguous. The set of integers has the same cardinality as the set of decimal places of .333… If you mean something different by size, you need to be explicit.