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September 17th, 2014, 08:26 AM   #11
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Are these ordinal number always whole numbers?
No. For example, neither $\omega$ nor $\omega^\omega$ are whole numbers.

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I am looking for a possibility of an infinitely small decimal number, infinitesimal, that is, it has a decimal part containing an infinite number of 0s and beyond this infinite amount of 0s there is 1. The number is 0.000....$\displaystyle \infty$......0001. Is it nonsense?
Yes, that's just nonsense.
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September 17th, 2014, 08:27 AM   #12
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Well, after you get to infinity, you have an infinite progression of transfinite cardinals: $\displaystyle \aleph_0$, $\displaystyle \aleph_1$, $\displaystyle \aleph_2$, etc. Of course these are all still "infinity". Any number greater than an infinite number would have to be infinite itself, so the question of what lies beyond infinity is kind of meaningless.
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September 17th, 2014, 08:33 AM   #13
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You might need to look at some formal theory of infinitesimals, which is definitely not mainstream mathematics.
I don't know about that, there are plenty of theories of infinitesimals which are well-known and fairly commonly used. It's just that the (nonzero*) infinitesimals aren't real numbers, and they don't have decimal expansions.

The hyperreals (nonstandard analysis) and the Conway-Knuth surreals are probably the most common. The so-called "dual numbers" might be the easiest to understand.

* Definitions differ on whether 0 is an infinitesimal. I think excluding it is most common.
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September 17th, 2014, 08:35 AM   #14
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Well, after you get to infinity, you have an infinite progression of transfinite cardinals: $\displaystyle \aleph_0$, $\displaystyle \aleph_1$, $\displaystyle \aleph_2$, etc.
Right. For any ordinal $\alpha$ there is a corresponding (much bigger!) cardinal $\aleph_\alpha$, so it's not just countably many alephs.
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September 17th, 2014, 10:53 AM   #15
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....there are plenty of theories of infinitesimals which are well-known and fairly commonly used. It's just that the (nonzero*) infinitesimals aren't real numbers, and they don't have decimal expansions.

* Definitions differ on whether 0 is an infinitesimal. I think excluding it is most common.
There are the limits $\displaystyle \lim_{x \to 0+} 1/x = +\infty$ and $\displaystyle \lim_{x \to 0-} 1/x = -\infty$

This suggests that there are three different zeros, +0, -0 and 0. Might the +0 and -0 be infinitesimals? Or how to understand this, perhaps there is an
infinitesimal difference between -0 and 0 and between 0 and +0.

I have arrived at this difference in this way: 1-0.999...... > 0 or
1 - 0.999........= +0
-1 + 0.999.......< 0 or -1 + 0.999......= -0
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September 17th, 2014, 11:02 AM   #16
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Originally Posted by TwoTwo View Post
There are the limits $\displaystyle \lim_{x \to 0+} 1/x = +\infty$ and $\displaystyle \lim_{x \to 0-} 1/x = -\infty$

This suggests that there are three different zeros, +0, -0 and 0.
No.

$0^+$ and $0^-$ are notational conventions that show the direction of approach to (the unique) zero. They are not different numbers.
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September 17th, 2014, 11:18 AM   #17
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This suggests that there are three different zeros, +0, -0 and 0.
In the real numbers all three are equal. In fact this is true in any ring, so the same holds in the rational numbers, the complex numbers, and the $p$-adics (for example).

Edit: Archie figured out why you were confused.
$$
\lim_{x\to0^+}
$$
just means that the path you take to get to 0 is from some point to its right. When you're taking limits in 2 or more dimensions you need a more complicated description of the path you take, but in 1 dimension all you need is a direction. Needless to say the path you take to get to a number is not the same as a number!

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September 17th, 2014, 12:12 PM   #18
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In the real numbers all three are equal. In fact this is true in any ring, so the same holds in the rational numbers, the complex numbers, and the $p$-adics (for example).
If there is an infinitesimal difference between +0 and 0 and -0 and 0
they don't need to be the same numbers. The problem is to calculate
what is this difference. My approach has been understanding the concept
of limit process.
I have managed to calculate the limits of single numbers. It may sound
nonsense, but I have noticed it works.
For example, the limit of 0 is not 0. So what is the limit of 0?
The limit of 0 is the infinitesimal, and vice versa: the limit of the infinitesimal is 0. It is possible to choose a symbol for the infinitesimal
so that it is easier to play with it, lets say it is x. Now we have: the limit of x
is not x, the limit of x is 0. I have noticed that the rule "the limit of A is not equal to A" applies always, not matter what is the number A.
Lately I found even very crazy developments: it is possible even to find the limits of the limits in this way. For example the limit of the limit of x is x.
The limit of the limit of 0 is 0.
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September 17th, 2014, 03:45 PM   #19
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My approach has been understanding the concept of limit process.
You have failed.

Your attempt to create a difference between the three ways of writing zero was to use $\sum \frac{9}{10^n} = 1$, as you would know if you understood limits. But we went through all that a few of months ago, which is why I ignored it before.

The rest of what you wrote is total nonsense. The limit of any constant $c$ as $x \to a$ is $c$.

I'd actuallly be interested in finding out about respectable theories of infinitesimals.
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Last edited by v8archie; September 17th, 2014 at 03:57 PM.
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September 18th, 2014, 05:53 AM   #20
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Quote:
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If there is an infinitesimal difference between +0 and 0 and -0 and 0
they don't need to be the same numbers.
In the real numbers they are all the same, and there are no infinitesimals (or just one, 0, depending on the definition you use).

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Originally Posted by TwoTwo View Post
I have managed to calculate the limits of single numbers. It may sound
nonsense, but I have noticed it works.
It's not nonsense. The limit of a constant function is always equal to the element in its range. For example, the limit of 1 is 1 and the limit of 0 is 0.

Your conclusions about limits are incorrect. If you write out a formal proof you should be able to find your mistake. I suspect that you could not write a formal proof, though, so the process is not as easy as I make it sound.
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