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July 4th, 2018, 09:02 AM   #1
pwi
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Paradox Reduced?

This ain't your father's decimal notation.

The new and improved interpretation of 0.999... (when the 9's go on forever).

People often say 0.999... has to be equal to 1 because there is no point between them.

I say that is true there is no point between them because they are points next to each other.

Think about it. If there is such a thing as "continuous", points have to be next to each other.

It is a given that two different rational points are never next to each other. I would say two different irrational points would never be next to each other either. That would leave rational and irrational points alternating if they could be next to each other. It does help to be able to think of infinite counting numbers.

So I say 1 is rational and 0.999... is irrational and are next to each other. I also would say any "decimal" notation can express any irrational number if it goes to infinite digits (giving infinite precision) while any "decimal" notation will leave out an infinite number of rational points (keep on reading to see that proved).

0.999.../3 = 0.333... correct? Everyone says that equals 1/3.

But I say 1/3 can not be expressed in decimal notation. If it could you would need 1/3 = A/10^B where A and B are finite integers and for decimal notation you need the denominator to be 10^B. If you cross multiply you get 3A=10^B but three will never go into 10 so the FUNCTION of converting 1/3 to decimal notation does not map all rational numbers to rationals (only the rationals that only have 2's and or 5's in the denominator can be represented). The decimal continuum is missing more rational points than it can represent.

It makes sense - an irrational number requires an infinite number of digits that are not zero (at its end, I would add if it is not an infinite counting number - I would say all points can be the ratio of two infinite counting numbers, some like 333.../999... can reduce to 1/3, or 10/3 or 1/30 ...) in order to be expressed.

An irrational is bounded above and below by a sequence of rationals, just fix the greater rational sequence at "one". So it would look like rationals and irrationals do alternate (like everyone sort of expected). So the measure of the rationals from zero to one is 1/2 or an infinitesimal above if both ends are included. Think of (N/2+1)/(N/2) for the ratio of rationals in [0,1] and (N/2-1)/(N/2) for the number of irrationals if N is the number of points in [0,1]. If this is true Measure Theory just became easier. Sorry.

So you can not look at the continuum with decimal notation. So use the slope of the lattice points and the origin. Notice if you do that and use the zig zag argument Cantor created you get a one to one with the counting numbers and the continuum and if you complete infinity to get irrational numbers you have a mythological list of the continuum.

Also look at (10^N-1)/10^N as N goes to infinity. Notice the numerator is always less than the denominator. Also notice that you get N 9's for any N. So if you try to say (10^N-1)/(10^N) = 1 if you multiply both sides by 10^N and reduce you get -1 = 0 a false statement but if you say (10^N-1)/(10^N) < 1 you get the true statement -1<0.

Once you know this everything makes sense. Cantor misused decimal notation. No more paradoxes (did anyone believe Banach Tarski was true? They did great work with what they had, but paradox tells you where to look for understanding). There is only one size of infinity.

Notice that Cantor required you to NOT be able to place a Power Set (defined as all possible subsets of a set) of an infinite set into a one to one correspondence with the counting numbers. Since the Power Set is related to the base two "decimal" continuum, let's do the "impossible" using base two "decimal" notation already shown to not be equal to the continuous real continuum. (Continuum Hypothesis come to mind?)

Let A={0.1, 0.01, 0.001, 0.0001, ... } in base two,
So the Power Set of A is equal to the base two "decimal" continuum if you just OR the subsets together. So look at the set B = {0.1, 0.01, 0.11, 0.001, 0.101, 0.011, 0.111, 0.0001, ...} where you read the mirror image of the element as a counting number associated with its place in the list. Example to get set B's 14th element use the base two counting number for fourteen = 1110. written backwards so 0.0111 and you can see you have to complete infinity to get any irrational number unless you just want to approach irrationals as close as you want forever.

Take a look at my books on Amazon (Duane Allred) for more proofs and consequences. The descriptions are interesting and you can peek inside for free. If you have Prime you should be able to read them for free.

So doesn't this make sense?

What are the best arguments you have that 0.999... is NOT next to 1? I say they always leave out the infinitesimal that is defined to not be equal to zero.

And remember the infinitesimal was made so limits could get as close as possible so as to not divide by zero, so why not go all the way the point next to where you would divide by zero?

Why can't points be next to each other? If they can't, the continuum would be pointless.

So what do you think?
Thanks from zylo

Last edited by skipjack; July 4th, 2018 at 11:21 AM.
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July 4th, 2018, 09:48 AM   #2
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Quote:
Originally Posted by pwi View Post
So what do you think?
We're not paid to think....
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July 4th, 2018, 10:31 AM   #3
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Quote:
Originally Posted by pwi View Post
If they can't the continuum would be pointless.
Pretty good pun!
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July 4th, 2018, 11:24 AM   #4
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So what is the average of 1 and 0.999999...?
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July 4th, 2018, 02:24 PM   #5
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Quote:
Originally Posted by pwi View Post
I say that is true there is no point between them because they are points next to each other.
Maths doesn't work on "what I say" or "what you think". It requires proof.

Quote:
Originally Posted by pwi View Post
If there is such a thing as "continuous", points have to be next to each other.
No. You are using finite reasoning for the infinite. Your intuition, like everybody else's, is badly flawed an doesn't know what it's talking about. This is why maths requires proof.

Quote:
Originally Posted by pwi View Post
That would leave rational and irrational points alternating if they could be next to each other.
But the cardinality of the irrationals is greater than that of the rationals. There is no 1-1 mapping between the two sets. So they don't alternate. It's a phrase that has no meaning anyway as there is no real number that is "next to" any other.

Quote:
Originally Posted by pwi View Post
What are the best arguments you have that 0.999... is NOT next to 1? I say they always leave out the infinitesimal that is defined to not be equal to zero.
The real numbers are the limits of convergent sequences of rational numbers. $0.999\ldots$ represents the series $\sum \frac9{10^n}$ and the limit of that sequence is $1$.

The real numbers do not include infinitesimals.
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