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Carleas wrote:James wrote:the most extreme member [of an infinite series]
Fault.James S Saint wrote:EVERY member of that set had to have a "0" as its right digit because you multiplied every member by 10.
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.
Uccisore wrote:James S Saint wrote:The "..." notation, specifically means that there is no end to be obtained. That means that it never, ever gets up to being exactly 1.0.
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.
James S Saint wrote:You are multiplying EACH member of the infinite set of elements. ... [T]hat requires that there be a "0" resulting from your multiplication that wasn't there before.
James S Saint wrote:Let's take this one step at a time.
To be correct, there can be no logical contradiction.A) Agree
B) Disagree
or Forfeit
Carleas wrote:James S Saint wrote:You are multiplying EACH member of the infinite set of elements. ... [T]hat requires that there be a "0" resulting from your multiplication that wasn't there before.
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.
James S Saint wrote:Let's take this one step at a time.
To be correct, there can be no logical contradiction.A) Agree
B) Disagree
or Forfeit
jerkey wrote:James S Saint wrote:Let's take this one step at a time.
To be correct, there can be no logical contradiction.A) Agree
B) Disagree
or Forfeit
b
James S Saint wrote: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.
James S Saint wrote:Last chance, Carleas:
Carleas wrote:James S Saint wrote: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.
Why? You can just shift the decimal point one space to the right. And you don't add a zero to the end because there is no end.
Carleas wrote:James S Saint wrote:Last chance, Carleas:
In the context of mathematics, A.
James S Saint wrote: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.
James S Saint wrote:What do you mean by "context of mathematics"?
Carleas wrote:James S Saint wrote: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.
Sounds good:
10(.333...) = 10 (.3 + .03 + .003 + .0003 + ...)
10(.333...) = 3 + .3 + .03 + .003 + ...)
10(.333...) = 3 +(.3 + .03 + .003 + .0003 + ...)
Turns out that was easy as fuck, not sure why you are having so much difficulty with it.
Carleas wrote:James S Saint wrote:What do you mean by "context of mathematics"?
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.
James S Saint wrote:Carleas wrote:James S Saint wrote:You are multiplying EACH member of the infinite set of elements. ... [T]hat requires that there be a "0" resulting from your multiplication that wasn't there before.
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.
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:James S Saint wrote:Let's take this one step at a time.
To be correct, there can be no logical contradiction.A) Agree
B) Disagree
or Forfeitjerkey wrote:James S Saint wrote:Let's take this one step at a time.
To be correct, there can be no logical contradiction.A) Agree
B) Disagree
or Forfeit
b
Please state your case for how a correct conclusion can be be drawn from logical contradiction.
jerkey wrote:Only through an intuition of math/reason, can a reductive differentiation of two types of logic manifest a connection.
James S Saint wrote:b]Next Question:[/b]
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.A) True
B) False
or Forfeit.
"quantity of digits to right of the decimal" ??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.
Carleas wrote:James S Saint wrote:b]Next Question:[/b]
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.A) True
B) False
or Forfeit.
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?
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.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.
phyllo wrote: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.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.
Carleas wrote: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.
Google wrote:car·di·nal·i·ty
ˌkärdəˈnalədē/
noun
Mathematics
noun: cardinality; plural noun: cardinalities
the number of elements in a set or other grouping, as a property of that grouping.
Merriam-Webster wrote:Definition of cardinality
plural cardinalities
: the number of elements in a given mathematical set
jerkey wrote: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.
James S Saint wrote:That is exactly what I mean by "cardinality" and by "size" - "the number of elements".
So what do YOU mean by those words?
B) NoDo you believe that there are different sizes of infinite?
A) Yes
B) No
or Forfeit
Of course, I am merely saying that I am naming one of the sizes.
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