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September 17th, 2014, 03:03 AM   #1
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what lies beyond infinity?

This is a very general question dealing with how do people usually deal
with the concept of infinity. It is true, for example, that there is no last digit of $\displaystyle \pi$, there is no last digit of unending decimal
number like 0.333.........

However, infinity seems to have a sign, it is either negative or positive.
There are $\displaystyle +\infty$ and $\displaystyle -\infty$.
These are not the same, for example $\displaystyle 0^\infty$ has
a limiting value 0 if $\displaystyle \infty = + \infty$, and 1/0 if $\displaystyle \infty = - \infty$
wikipedia link

It is possible to think that after writing infinity, also infinite amount of digits, we need still write down something more, either + or -. Therefore something seems to "lie beyond infinity".
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September 17th, 2014, 03:42 AM   #2
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The is no $\infty$, it is a notational convention meaning that a variable is allowed to grow without bound. The concept "beyond $\infty$" has no meaning.

Symbollically, $\infty$ is usually equivalent to $+\infty$, but on occasion is used when we do not yet know which of $-\infty$ and $\infty$ are appropriate. This is usually because we are talking generally enough that our arguments cover both cases.

$0^\infty$ is an indeterminate form. It can take any value in the limit, but has no value of it's own. Your link talks about $\frac{1}{0}$ which is not indeterminate, it simply is undefined.
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Last edited by v8archie; September 17th, 2014 at 03:52 AM.
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September 17th, 2014, 04:25 AM   #3
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Quote:
Originally Posted by v8archie View Post
The is no $\infty$, it is a notational convention meaning that a variable is allowed to grow without bound.
In calculus, maybe. In other areas of mathematics infinity can be defined, for example, in projective geometry and ordinal numbers.
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September 17th, 2014, 05:48 AM   #4
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There is no single concept of "infinity". In fact, it's a good general sign that you're not dealing with mathematics when someone says "infinity". (N.B., the adjective "infinite" is quite common in math.)

There are things called $+\infty$ and $-\infty$ in the extended real line, and they do work more-or-less like you'd expect from the "calculus $\infty$" which is, as v8archie says, just a convention rather than an object. In the extended real line there is nothing larger than $+\infty$ or less than $-\infty$ so the answer to your question here is "nothing".

There is also an object called $\infty$ in projective geometry which is neither positive nor negative. As a result it doesn't make sense to ask what is beyond.

But you mention the decimal digit of a number, so maybe we'd do better using ordinals. Ordinals are defined in set theory and use it pretty extensively. Let's start with a set-theoretic definition of 0 as the empty set {}. Then define n+1 as {0, ..., n}. An ordinal is just an extension of this definition: it is a set containing all smaller ordinals, with the understanding that 0 = {} is the smallest ordinal. You can always get the next ordinal after $\alpha$, let's call it $\alpha+1$, by taking $\alpha\cup\{\alpha\}.$

The smallest infinite ordinal is called $\omega$ and it is defined as the set {0, 1, 2, ...} containing all nonnegative integers. (In other contexts you'd call this set the natural numbers but here it's just called $\omega$.) A real number has $\omega$ decimal digits after the decimal point which makes sense -- you can number each of the decimal places, there is no decimal place without a number (or between numbers).

So is there an ordinal beyond $\omega$? Yes! It's called $\omega+1$ and it is defined as $\omega\cup\{\omega\}=\{0,1,2,\ldots,\omega\}.$ In fact there are infinitely many ordinals larger than $\omega,$ like $\omega+3$ and $\omega^{\omega+1}+\omega^7\cdot4+2.$

So the answer to your question is "yes", "no", or "your question doesn't make sense" depending on how you interpret it. The world of mathematics is large, and I haven't even scratched the surface here...!
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September 17th, 2014, 07:28 AM   #5
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Originally Posted by Evgeny.Makarov View Post
In calculus, maybe. In other areas of mathematics infinity can be defined, for example, in projective geometry and ordinal numbers.
I wasn't aware of the infinity in Projective Geometry, but the ordinals are not represented using $\infty$, so it seemed reasonable to suppose that we weren't talking about those.

Besides, CRG knows more about those than I ever will, and this is the sort of thread that would attract him - witness his post above.
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September 17th, 2014, 07:52 AM   #6
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I wasn't aware of the infinity in Projective Geometry
You don't have to call this infinity, and it can be given a label which uses only finite numbers.

A point on the real projective line can be defined as an equivalence class of ordered pairs (x, y) with x and y real, where (x1, y1) and (x2, y2) are identified if and only if there is some nonzero k with x1 = k*x2 and y1 = k*y2. The special point (1, 0) is usually called the "point at infinity" because, unlike the other points, it's not equal to some point of the form (x, 1).

There are good reasons for calling these things infinite, but it's all a matter of perspective. To me that formulation makes the most sense and the ordered pair definition is awkward. But it's just a matter of taste -- the mathematics is the same either way.

If you move up to the real projective plane then instead of a point at infinity you get a line (and a point) at infinity.
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September 17th, 2014, 08:02 AM   #7
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Quote:
Originally Posted by CRGreathouse View Post
There is no single concept of "infinity". In fact, it's a good general sign that you're not dealing with mathematics when someone says "infinity". (N.B., the adjective "infinite" is quite common in math.)

There are things called $+\infty$ and $-\infty$ in the extended real line, and they do work more-or-less like you'd expect from the "calculus $\infty$" which is, as v8archie says, just a convention rather than an object. In the extended real line there is nothing larger than $+\infty$ or less than $-\infty$ so the answer to your question here is "nothing".

There is also an object called $\infty$ in projective geometry which is neither positive nor negative. As a result it doesn't make sense to ask what is beyond.
I think that there are at least two different infinities, positive and negative.
Maybe also a "neutral" infinity $\displaystyle \infty$ containing both positive
and negative infinities, similar to a neutron which decays into a positive and
negative particles, a proton and electron.

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Originally Posted by CRGreathouse View Post
The smallest infinite ordinal is called $\omega$ and it is defined as the set {0, 1, 2, ...} containing all nonnegative integers. (In other contexts you'd call this set the natural numbers but here it's just called $\omega$.) A real number has $\omega$ decimal digits after the decimal point which makes sense -- you can number each of the decimal places, there is no decimal place without a number (or between numbers).

So is there an ordinal beyond $\omega$? Yes! It's called $\omega+1$ and it is defined as $\omega\cup\{\omega\}=\{0,1,2,\ldots,\omega\}.$ In fact there are infinitely many ordinals larger than $\omega,$ like $\omega+3$ and $\omega^{\omega+1}+\omega^7\cdot4+2.$
Are these ordinal number always whole numbers?
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?
If this number is written from right to left, it is
1000....$\displaystyle \infty$.......000.0
In this representation the 1 appears in front of infinite amount of digits,
just like + and - of the $\displaystyle +\infty$ and $\displaystyle -\infty$.
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September 17th, 2014, 08:04 AM   #8
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Oops! I mis-spoke about $0^\infty$. It is not indeterminate, but neither does it have a value.
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September 17th, 2014, 08:11 AM   #9
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The different "infinities" in are not different in the way you are considering them.
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Originally Posted by TwoTwo View Post
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?
In short, yes. Your number is
$$\sum_{m=0}^\infty \frac{0}{10^m} + \lim_{n \to \infty} \frac{1}{10^n} = 0 + 0 = 0$$

You might need to look at some formal theory of infinitesimals, which is definitely not mainstream mathematics.
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September 17th, 2014, 08:24 AM   #10
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Quote:
Originally Posted by TwoTwo View Post
I think that there are at least two different infinities, positive and negative.
Maybe also a "neutral" infinity $\displaystyle \infty$ containing both positive
and negative infinities, similar to a neutron which decays into a positive and
negative particles, a proton and electron.
It depends entirely on the system you're working in. Your comparison is inapt.

In the real numbers, the integers, the rational numbers, and the complex numbers there are no infinite elements at all.

In the projective reals, like the Riemann sphere, there is exactly one infinite element.

In the extended reals there are exactly 2 infinite elements.

In the real projective plane there are $\beth_1$ infinite elements: infinitely many, but not 'too big' of an infinity.

There are infinitely many infinite ordinals, and the number of infinite ordinals is larger than $\beth_1$ or indeed any infinite cardinal. There are a lot of infinite ordinals!
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