No matter how many 3s are implied by the ellipsis, the expression is never equal to (\frac{1}{3}). In other words, even with a (3) located at the position with an index of (-infA) (or any other negative infinity), the number would still be less than (\frac{1}{3}).
The larger the number of (3)s, the closer the number is to (\frac{1}{3}). But it can never be exactly (\frac{1}{3}).
That is basically contradicted by all the mathematical results shown in this thread.
Sure, you can say that but at the same time you are saying that the sum of a converging geometric series is wrong. (For example)
Which result do you prefer to retain?
I prefer not to throw away the work on geometric series.
Do you think that you can Resolution Debate that proposal?
No problem. But let’s not start a new debate until we review and summarize our first debate. Also, I’d want to make certain things clear in advance. (And I won’t be available until the weekend or so.)
But I’m surprised you disagree. What’s your position on the issue?
I think you’re presenting a false dichotomy. You either accept that (0.33\dot3 < \frac{1}{3}) and throw away the work on geometric series or you reject that (0.33\dot3 < \frac{1}{3}) and keep the work on geometric series. In reality, you can accept that the two numbers aren’t equal AND keep the existing work by making an adjustment to it.
What they have shown is that the limit of (0.3 + 0.03 + 0.003 + \cdots) is (\frac{1}{3}). I don’t dispute that. But then they erroneously conclude that (0.33\dot3 = \frac{1}{3}). It’s like someone correctly showing that the ceiling of (1 + 5 \div 10) is (2) but then erroneously concluding that (1 + 5 \div 10 = 1).
But that is the DEFINITION of the sum of the infinite series.
It’s not an erroneous equating. It’s a definition.
What’s the point of repeating this point? I’ve already responded to it. Why not address my response instead?
I am fully aware that’s how they define it. The thing is that that definition conflicts with existing definitions.
If we define symbols “2” and “+” the way we normally define them, then the meaning of “2 + 2” can be deduced to be the same as the meaning of “4”. You are thus not free to declare that “2 + 2” means “10” and not “4”.
“0.3 + 0.03 + 0.003 + …” has its own meaning and someone (whoever that is) declaring that it means something else is introducing a contradiction in their system of thought.
That expression DOES NOT stand for a limit.
(In the same exact way that “1 + 2 / 4” does not stand for the ceiling of “1 + 2 / 4”.)
It does stand for a limit, that’s the ONLY thing it can stand for.
Once you agree that the limit of .3, .33, .333, … is 1/3, we’re done.
I truly don’t understand your point. Addition is a binary operation. The only way to define addition of infinitely many summands is to define it as the limit of the sequence of partial sums. Having done that, we’re done. You agree to all this … then you say no. This I truly don’t get.
Who said I disagree?
So who would you want to debate?
I can moderate.
Look up the definition of the sum of an infinite series. It’s defined as the limit of the sequence of partial sums.
Secondly, you posted a handwavy Wiki definition of limit, not the technical definition. What is the point of flaunting your lack of mathematical understanding? If you don’t know what a limit is, copypasting Wiki won’t help.
A series sum is a function as long as there are variables for the limits (which there are).
They say that the limit is “called” the sum. That admits that it is actually a limit and that it is merely “called the sum”.
Add to that -
If you don’t like Wikipedia, show us a better source.
Or better yet - go talk to them and correct them. 
Then your ‘=’ must be taken to mean near to but not an exact match to, or, nearly equally to but not truly equal to.
I’m looking at this from a non-mathematical point of view. I don’t think you can have multiple infinitesimals. To me, infinitesimal is that which separates all possible measures and things from each other. To me, only one existing thing can take the measure of infinitesimal and infinite. I call It Existence/God.
Yes, addition is a binary operation. It is a function that takes two values and outputs a single value. We know that. Do you have anything else to say? Because you are repeating yourself.
You seem to think that an expression such as “2 + 2 + 2” represents an operation of addition that takes three values and outputs a single value. If that’s what the expression stands for then it contradicts the definition of the word “addition”. As you say, addition takes two inputs; but this expression, you also claim, represents a function that takes three. How can it then be addition? It cannot be – unless you change the definition of the word “addition” that is. Fortunately for us, that’s not what the expression stands for. It stands for a chained series of additions where the result of the first addition (which is “2 + 2”) is taken as the input of the second addition. This means that you DO NOT have to redefine the concept of addition. You DO NOT have to extend it so that it can take three (rather than merely two) values as its input.
Exactly what I explained to you in detail a couple of posts back.
Yes you STILL have no definition of an infinite sum, since you can only use induction to reduce a finite sum to a succession of binary operations.
Well folks it’s been fun, I’m done here. All the best.
I can debate anyone: Certainly Real, phyllo, wtf, Ecmandu, etc. But we’ll have to find a set of rules that both me and the other participant are willing to abide by. The job of a moderator (you or anyone else) would be merely to check that the rules are respected and to intercept when violation is suspected.
No one can agree to rules until they hear them. Why don’t you propose some on the General Resolution Debate thread. We can talk about it.
Hi all, thought I’d contribute the following to this thread.
A) The set of all numbers (if it’s a non-contradictory number, then it is included in this set)
B) The set of all numbers except the number 10
It seems that both A and B encompass an endless number of numbers. However, one cannot deny that A is greater than B in terms of quantity. I will attempt to show: 1) B is semi-infinite in quantity, whilst A is infinite in quantity, and 2) Semi-infinites come in various sizes, but there is only one infinity (so there aren’t infinities of various sizes).
If you tell me “there is no end to the number of numbers that B encompasses”, and I ask you “does B encompass the number 10?”, you will say “no”. To which I will say “clearly, there is an end to the number of numbers that B encompasses. Had you said ‘excluding 10, there is no end to the number of numbers that B encompasses’ I might have believed you”. I say might because I’m not sure if B encompasses infinity. Either we say A is infinity (in which case B does not encompass infinity), or we say A is not infinity (in which case both A and B encompass infinity, but only A encompasses/is an infinity of numbers).
Whilst there absolutely/truly is no end to the number of numbers that A encompasses, there is an end to the number of numbers that B encompasses in an absolute sense. Having said that, the number of numbers that B encompasses is not finite in quantity (hence the term semi-infinite). Furthermore, B is one possible maximally large semi-infinite set of numbers (because it encompasses all numbers but one, and there are an endless number of semi-infinite sets that do this. A semi-infinite set that encompasses all numbers but two is smaller than the aforementioned semi-infinite set).
Hopefully, the above proves that whilst there are many semi-infinite sets of varying sizes, there is only one infinite set of numbers.
Umm…
CR,
What you posted is an extreme lack of understanding of correspondence. Cantor figured out correspondence a couple hundred years ago.
Except for your use of the word “infinity” (instead of “infinite”) I can accept all of that but -
Wouldn’t the set of two truly infinite sets be “larger” than merely one infinite set?