Math Is Fun Forum

Great! This is what I have been waiting for, Ricky. And without explicitly using the concept of infinity I still can figure out the dilema.

Not finite in number? This stating is a step from perfect defination- Let us define it as not only nonfinite in number but also LARGER than any number, and it has to be so otherwise 9's  less than any finite 9's obviously cannot defeat sufficient 9's after the digit point   begining the interval [x,1]. And it is nothing wrong to name the quantity of 9's as a superquantity q that has the property q>N, in which N is any integer, is it?

However, what does the 9 after q-1 9's represent? You won't say this is invalid, will you? For if it is, not finite 9's is even not possible or feasible. Back to the topic, this 9 represents a quantity 1/10[sup]q[/sup] because it is situated as the qth digit from the digit point. You wouldn't say it does not represent so, would you? (If you would, please tell me what does it represent- a 0 is only fine.)

And this 1/10[sup]q[/sup] is conceivably less than any positive rationals(Why I use ratioanals instead of quantities will be illustrated later. However quantities will  equally function here). For  any rational α, there will be at least one N to insure that 1/10[sup]N[/sup]<α. Hence 1/10[sup]q[/sup] would be less than α too. What? You want to say that it isn't- do you want to state that this 9 represents sth larger than some 9 before it? Thereby it still represents a quantitiless. (It is only fine if you name quantitiless, instead, 0)

The same dilema again, you can forbid the expression of q/1,000,000,000, but you will still be unable to figure out from where 9's start to represent quantitilesses.

By the way, just by stating infinity is not finity doesn't solve any problem. Logically infinity and finity are completementory and together exhaustive. Stating "not finite" is the same as stating some "quantity" "infinite", as I abbreviated as q(you can use any other symbol but that would change nothing essential).

Further, if you deny this, you have an attitude problem I am afraid to say. Not to me, though, but to the concept of "infinite 9's" you've employed. In the "proof" on post 76, you employed this guy to outweigh any finite 9's after the digit point that starts the intervals like  [0.9,1], [0.99,1], [0.999,1] etc. But now when you face the dilema created by it, you fire it, and depose of it . If you haven't got it, reread from Paragraph 2.

It is a 0. and then an infinite number of 9's after it.

So you have interpreted 0.999...

Here, you have used infinite 9s , and it wouldn't be appropriate for me to further it by there are infinite digits after the digit point, would it?

Well, now what does the 9 on Digit Infinite, or the Infiniteth digit stand for?

Does it stand for nonsense?
No

Does it stand for any finite quantity?
No

Does it stand for a 0, or an infinitesimal or a quantitiless?
Yes, reasonably. Because literally it stands for

Now, let's say it is 0 because you once said so.

The digit prior to it represents  also 0. When it does not, the digit after it will represent one tenth of it.

The digit N digits prior to the Infiniteth digit is 0, too, no matter how large the N is.

Let us see what the 1/2∞th digit stands for. It must be 0, because if it is any x>0, Digit ∞ will stand for sth in the x^2 scale.

So it is ok to say that Digit

Hence what does 0.999... mean now?

From left, each 9 stands for a positive quatity, no matter how small but not equal to 0, from the right, or from middle-anywhere after ∞-1 9s,  each 9 stands for 0, please do tell me when 9 starts to represent 0?

533+1/3 at most

I proved 533+1/3 is the most in my home forum. But I lack time to translate it into English currently.

Stangely enough, someone did say 567 could be an answer at that time.

Dross refuted my proof that the infiniteth 9 is unreachable by stating sth like" imagine you add all of them together". So long as he had said the "all", I found it appropriate and necessary to imagine the end in the "all" as well, which was a temp concilatory and did not mean I gave up my stand.

Quatum mechanics is so popular only because it is the mere case that defy causality. The nuclear more or less obeies the causality and stay more or less the same at where it were.

Archimedian principle and completeness- I will challenge this principle quite soon. Sorry I haven't aquired the knowledge of interval proof, but they should be  collarabotively false with Archimedian principle equalavent to Reals completeness.

Finite logic is addition, substraction, multiplication, division, mathematical deduction and Cauchy's defination of limit(e-N or Delta system), particularly when used in trying to prove 0.999... equates 1, or to prove 0.111...=1/9.

Can either of you prove either of the equities using those method? But keeping in mind: Don't make a defination whenever you Fail to prove.

On Ricky's words in Post 80, you haven't explicitly pointed out HOW I will tear the base of Calculus down-or put in other words, you haven't supported your assertion, so in no way  I can refute or reply you. But don't think in this way you are right because you have not prove the assertion yet.

Then comes the more convincing argument: that the last 9 is quantitiless, infinitesimal that equals to 0, or quasi-0. 

It, however, is also doubtful. The basic doubt is that when and how it is gotten? Zeno used to question the quantitiless concept by analyzing a moving arrow:
The arrow should  reach the mid point before reaching the end, but should also reach the quater point before reaching the mid point,... Each step is finite, and it seems no hope to get an infintesimal or a quantitiless by reducing the quantity, in number.

Still, let us say somehow the quantitiless or the infinitesimal is reached, then how about add the sum up this way? -From the last, backwards.

the last 9 is quantitiless, the second last one is also quantitiless, the N last one is too.

Going further, we can say that the half-infiniteth 9 is quantitiless- It has to be otherwise the last 9 is not. (think of any digit number then double the collective 0s after the digit point)
Even Entry 1/1,000,000,000 of infinity should be a quantitiless.

And when does the entry turn out to be a quantity from a quantitiless? Which entry?
This is the reverse question of when  quantities come to a quantitless state.

Altogether, can anyone distinguish which entry separate the quantities on the left and the quantitiless on the right? Or, can anyone illustrate how the quantities grow to the quantitiless gradually?

To be continued...

Until now in this topic,  having debated so long, Ricky, Dross and I all admit that:
1) 0.999...=0.9+0.09+0.009+...;
2) Either 0.999...=1 or 0.111...=1/9 cannot be proven by any finite logic, and either a defination or an imagination is needed.

And Dross insists the imagination or intuition method, while Ricky favors a defination he thinks useful and essential. I, certainly, deny both.

One day, Dross found a long passage written by a professor on wikipedia, which tries to prove 0.999...=1. Since it looked a big deal and I lacked time, I asked for a pause and promised to challange the 0.999...=1 thing later.

Now it's been too long for a pause, thus I complete the challenge below.

Forever young......
-A song by an English female singer

You are welcome