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.
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”.
I agree with you. But I want to be charitable and allow extended reals so that we can call the limit of 1, 10, 100, 1000 … (\infty). But @Magnus will not even agree to that. So I’m at a loss.
Great. Then tell me what (10^\infty) means “according to the standard mathematical definitions,” and please make sure to reference those definitions. Because frankly I’m highly familiar with “standard mathematical definitions,” and you have not presented one yet.
Yes, but the question is can a circle be divided into 3 wholly equal parts? I know I can divide a circle into 3 non-equal parts. I am not certain if I can divide it into three wholly equal parts. I think it’s doable, and given pi, this is supposed to be doable. As far as I can see, if you accept pi, then consistency would have it that you accept 1/3 to = 0.333…
I don’t see how we can say that pi only contains a finite amount of information. I understand we can represent this infinite amount of information in a finite manner. But this is not the same as pi itself containing a finite amount of information. If pi consists of an infinity of digits, then that is an infinite amount of information because each number is a bit of information.
Where we can divide a circle into 3 wholly equal parts using pi, then pi is infinite. 1/3 is a finite part of something. Pi is more like a tool to do something with than a finite part of something.
I meant that it says that it consists of an infinity of digits.
See my above point on the representation of pi versus the value of pi.
I mean no offence by the following, just constructive feedback: I think you are too focused on representation and not focused enough on semantics.
I agree that you can get as close as you want. We were never in disagreement on this. And that was my point to you when you first said 1/3 = 0.333…
If you remember, I said that 1/3 cannot equal .333… because an infinity of 3s are impossible and an infinity of 3s are needed to fulfil the semantic of 1/3 as opposed to just get close to it. But then you made that point about circles and I reconsidered my position.
To emphasise, you say you do not need to talk about infinitely many 3s, but then semantically speaking you have not fulfilled the semantic of 1/3 have you? When the 3s are finite, you have only fulfilled close to 1/3. The only way you can say 1/3 = .333… is if you are saying that an infinity of 3s follow.
Certainly - 120° each slice = exactly 1/3 of a circle.
Still - you are confusing the digits involved with the value they represent. A very small number might be represented by an infinity of digits - that does NOTmake the number infinite. And it does not stop anyone from dividing something into 3 equal parts.
The decimal system of listing numbers has issues - problems. Some values (such as 1/3) cannot be entirely and precisely represented with decimal numbers. The issue is only in the language being used (base_10 digits) - not with the real values.
Pi is a value that cannot be exactly represented by decimal digits ever. The value of Pi doesn’t care whether you try to represent it in decimal form. The value of it doesn’t change to match your estimations. The value is a little over 3.14 and always will be even if you list an infinity more digits in the effort to be 100% exact. Pi will always just be a little over 3.14.
Similar with 1/3. It doesn’t matter how many digits your want to throw together to try to represent it. It will always remain merely 1/3 or a little over 0.33. Add a billion more 3’s at the end and you do not change what 1/3 is at all. You only change the accuracy of your string of decimals.
The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.
The language keeps saying that 1/3=0.333… and 1=0.999… through series convergence and the many other techniques that have been demonstrated in this thread.
Take the semantic of ‘21’ and the semantic of ‘divide’ and the semantic of ‘3’ and the semantic of ‘equals’. What does 21 divided by 3 equal? The answer is not 21/3 because that is the question that is being asked. It is not the answer. The answer is the semantic of 7. 7 is the answer in its actual purest form.
Now, what is 1 divide by 3 equal to? The answer is either absurd or .333… (where … stands for an infinity of 3s)
You cannot say the decimal system of listing numbers has issues. It is the semantic of ‘one’ and the semantic of ‘three’ and the semantic of ‘divided by’ such that when you ask what is one divide by 3? the answer is either you cannot divide 1 by 3 or it is zero point followed by an infinity of 3s.
Yes but semantics dictate that either 1/3 is absurd, or 1/3 is 0.333…
It is semantically inconsistent to say that 1 divide by 3 is 0.333… (follow by a vary large but finite number of 3s).
I see them as having the same identical answer. The semantic of 7. I don’t see them as being semantically identical. As in 21/3 is not semantically identical to 7/1. Their answers are semantically identical. This does not make them semantically identical.
It might help us (and you) if you would describe what certain words mean to you - define them in your own words - such as “semantics”, “infinite”, “…”, and “infinity”.
I don’t have any doubt that this whole thread is nothing but arguments over words and symbols.