I don’t think that “10 multiplied by itself infinitely” is undefined. And I also do not think the rest of your post shows otherwise. What makes you think that an infinite number of numbers multiplied together cannot be reduced to an infinite sequence of binary operations? And what do you want me to do to prove that an infinite product such as (10^{\infty}) is not undefined?
It is an ambiguous term. You don’t know if it means merely the size of the natural numbers (such as (infA)) or 10 times that or that squared. If you try to use it in maths, you find that you can prove almost anything is equal to almost anything else.
(\infty + 10 = \infty + 3 = \infty^2 = \infty - 40)
Because what is infinite is infinite - especially if you add anything to it.
I thought I already did.
The ellipsis is to the right side of the digits which indicates that the digits are expanding to the right, which is to say, in the direction of the least significant digit.
Let (x) be the index of the first (the leftmost) (9) in (99\dot9). With that in mind, the index of the second is (x - 1), the index of the third is (x - 2), the index of the fourth is (x - 3) and so on. And since the ellipsis indicates that this process continues without an end, there is no last digit, which means, there is no (9) with the lowest index (“the least significant (9)”).
In the case of YOUR number, which is (\sum_{i=0}^{i->\infty} 9\times10^i), the least significant (9) exists.
That alone tells you there is a difference between the two numbers.
If it’s allowed to go below zero, the concept becomes contradictory, which means, it cannot stand for anything real, but that does not mean we can’t talk about it and/or use it in other ways.
And note that I didn’t say that (999\dotso) necessarily represents that concept. It can be used to represent that concept but it can also be used to represent other concepts. It all depends on the index of the first digit. If the index of the first digit is (infA^2), and the number of (9)s is (infA), then “…” does not extend beyond the decimal point.
Normally, decimal numerals have “the rightmost digit before the decimal point”. (150) is an example. The rightmost digit before the decimal point is (0). In such a case, we have a rule for determining the index of any digit within the numeral. The index of the rightmost digit, for example, is (0). The index of any other digit is based on how far away it is from the rightmost digit and whether it is to the right or to the left of it. For example, the index of (5) is (1) and the index of (1) is (2).
But (999\dotso) does not have “the rightmost digit”, so we cannot use this rule. And since no other rule exists, there’s no way to deduce the index of any digit.
You can do logic with ambiguous terms – if you know how to do it. And you can’t prove just about anything with (\infty) unless you’re using it improperly.
You merely repeated what I had already rebutted. Repetition doesn’t count as a rebuttal - it indicates that there is no more argument to be presented and the end of that encounter.
Again, you are repeating yourself. I already explained why that is not acceptable -
I see that I misspoke - I meant to say that it cannot go below the decimal point.
It cannot because of the reasons given above.
Again you are repeating what I have already explained away - your number cannot be of a higher cardinality than the natural numbers because that is all we are dealing with and the ellipsis is only about the natural “countable” number set.
That is counter to how you were indexing originally but I agree that an index should begin AT the decimal point - and that means that your first “9” of your “999…” figure is AT index (infA) (or (infA-1) if you prefer), else the entire number doesn’t make sense.
I explained that “other rule” in my first rebuttal in our first attempt -
Where is it defined? If you have defined it, I missed it, so please repeat the definition. But if you haven’t defined it, then where is it defined? There is of course a notion of an infinite product in math, but it has a technical definition, and your usage doesn’t fit that case. At best we can say that your notation is an infinite product that diverges. If that’s what you mean, I’ll accept that. I am not sure though that this is what you mean.
Magnus prefers - “if you claim that it is not there - you should prove that it is not”. :-"
Been through that with him already. 
I don’t think I repeated myself. I stated something I never stated before – what I mean by “implicit digit”. I thought you did not understand the term.
You seem to think that if a meaning is contradictory that it cannot possibly be associated with a word.
It’s like saying the meaning of “square-circle” is not “a shape that is both a circle and a shape” because a shape cannot be both a circle and a shape.
I understood you correctly.
Yes, that’s what you think. I disagree with that.
The only difference is that I’m starting with (0).
That’s how I did it before:
ilovephilosophy.com/viewtop … 5#p2803118
That does not follow. That’s merely you stitching things together so that they make sense to you. That’s like me saying “square-circle” means “square” because otherwise the concept is contradictory. You don’t get to decide the meaning of an expression based on what makes sense to you and what doesn’t. It means what it means.
The onus of proof is ALWAYS on the one making the claim (whether that claim is positive or negative is irrelevant.)
“10 multiplied by itself infinitely” is another expression for (10^{\infty}).
And if you want me to define the entire expression, I don’t think that’s how it works. The meaning of a statement is derived from the meaning of its words (among other things.)
So what exactly do you want me to define?
What exactly is unclear?
(10^{\infty}) is undefined, it’s not a well-formed formula in any mathematical system I know. Have you looked at the definition of an infinite product I linked? At best you could define (10^{\infty}) as (\displaystyle \lim_{n \to \infty} \Pi_{k =0}^n 10^k), but that infinite product diverges. That is, it’s the limit of 1, 10, 100, 1000, 10000, …
If you want to call it (\infty) in the extended real numbers you could do that. But it doesn’t mean anything significant. Certainly not the way you’re trying to use it.
They define infinite product as the limit of partial products. That’s a different concept.
I understood the first time. Did you understand that what I said was that I did not see the implication of the 0 - I didn’t say that I merely did not see the 0.
“is contradictory”? If the meaning of a word is undefined (or worse self-contradictory) then the word is ambiguous and cannot be used to prove anything. To prove is to remove doubt, question, or alternative.
Except that “a number” was presented - not an oxymoron. If you want to claim that the “999…” isn’t really a number that is a different argument.
Then why haven’t you simply said “I disagree that it must be within the natural number set”?
We could go from there.
If you want to claim that the “999…” is an oxymoron then that constitutes a definitional premise disagreement and is in opposition to the implied definition of my proposal - because in my proposal it is NOT meant as an oxymoron.
You are not allowed to change definitions intended by the OP although asking for them or clarifying them is certainly permitted. Like I said - finding agreement is the issue - not trying to win an argument.
You said there is no implicit (0) that comes before the leftmost digit in (99\dot9).
And you reasoning was:
In (99\dot9), the three dots (“…”) say no more than “repeat (9) endlessly in the direction of the least significant digit”. They say NOTHING about the digits that comes before the leftmost (9).
And if the index of the first (9) is (infA), they are also not saying anything about the digit associated with (10^0) (the (9)s do not extend enough to reach that point.)
Right, so if it’s a number, it’s not an oxymoron. But if it’s a shape, then it might be an oxymoron (:
My task is merely to prove that (999\dotso) is not (\sum_{i=0}^{i->{\infty}} 9\times10^0).
I think I said that long time ago. (But not during the Resolution Debate we started.)
We’re trying to DEDUCE the meaning of (999\dotso) using existing mathematical definitions, not merely invent a new one.
If your claim is merely “This is how I define (999\dotso)” then there is nothing to debate.
Ok then you are arguing with the definition intended in the OP.
Since I am the one who presented that proposal - my intended definition was that “999…” is an infinite quantity in the set of natural numbers - with no implicit digits to the left of the leftmost 9 nor to the right of any decimal point."
End of game.
Unless you have an argument against what you now understand the original post to mean.
That’s how it’s defined.
If you have a different definition: please supply it forthwith.
You haven’t done so. And now you say you are using existing mathematical definitions, but you just rejected the ONLY existing mathematical definition of an infinite product, which is the limit of partial products. And which DOES happen to make sense in this case, except that it trivially diverges to infinity.
So please show me your definition of (10^\infty) that does NOT involve the limit of partial products; but DOES rest on existing mathematical definitions.
I don’t exactly remember on which page but at one point you made a claim that the meaning of (99\dot9) according to standard mathematical definitions (not according to your own definitions) is (\sum_{i=0}^{i->{\infty}} 9\times10^i). It is this claim that I suggested be the subject of ILP’s first Resolution Debate ever (remember that it was me who suggested we debate this subject.) But now, you’re trying to bail out on the ground that your proposal merely spoke of your own personal definition ): I mean, anyone can define words any way they want, and unless there’s a reason to think they are lying about the way they are defining them, there is no reason to disagree with their definitions. It’s a category of beliefs least suitable for a debate (since the point of debates is to resolve disagreements.)
I rejected a definition that mathematicians developed because they didn’t know how to properly deduce the meaning of certain category of expressions from existing definitions.
An analogy would be a man who due to his inability to calculate the result of “2 + 2” says “Fuck it, let’s just define it to be 10!”
When I say that an infinite product is this or that, I am not merely saying “This is a personal definition of mine”. I am actually saying “This is what the term means according to the standard mathematical definitions”.
If you think you are going by “standard definitions” then we disagree on that too.
That could only apply to convergent partials.
The limit of divergent partial products would remain undefined.