Full-fledged arithmetic with zero and infinity

Psalehesost

Psalehesost

The Living Force
I have written thoughts on this subject from time to time gathering virtual dust on my hard drive for half a year.

I just made a more clear and concise writeup which I will now post.

My guess is that this is either very clever or very stupid, and will either give a good laugh or a good thought to a true mathematician (which I'm not - so please excuse the likely somewhat deficient language).

No idea which.

Code: 
Full-fledged arithmetic with zero and infinity
----------------------------------------------

	By <real name removed for now>

Symbol substitutions in the ASCII original of the document:
	"Inf" substitutes the infinity symbol.

Basic definitions:
	"Order of Infinity": If higher in x than y, x will be as infinity
	to y, and y will be as zero to x. If lower in x than y, then the
	converse.

	"Zero" (0) by itself is, in ordinary circumstances (when 0 does
	not represent a small but unknown quantity), the same as 1 of
	the order -1 of infinity.

	"Infinity" (Inf) by itself is, in ordinary circumstances (when Inf
	does not represent a large but unknown quantity), the same as 1 of
	the order 1 of infinity.

	"One" (1) and all other non-zero and non-infinite multiples thereof
	are of the order 0 of infinity.

Operations:
	x+y gives the sum of the values if of equal order of infinity,
	otherwise the value with the highest order of infinity. Order of
	infinity is unaffected.

	x-y gives the difference of the values if of equal order of infinity,
	otherwise the value of x or -y having the highest order of infinity.
	Order of infinity is unaffected unless the result is 0, in which
	case it has decreased by one.

	x*y gives the product with an order of infinity the sum of the
	orders of infinity of x and y.

	x/y gives the quotient with an order of infinity the difference
	of the orders of infinity of x and y.

From this the simple cases follow:
	x+0 and x-0 give x.

	x+Inf gives Inf, x-Inf gives -Inf.

	x*Inf and x/0 give x of an order of infinity one greater.

	x/Inf and x*0 give x of an order of infinity one less.

Some implications:
	0/0 = 1
	Inf/Inf = 1
	(x*0)/0 = (x/Inf)*Inf = x
	(x/0)*0 = (x*Inf)/Inf = x
 
Building upon the above, I continue the mathematical thought-experimentation inspired by some reading of Ouspensky's Tertium Oraganum:

Ouspensky gives the example of a line, regardless of length, having an infinite number of points. I would consequently say that its magnitude has an order of infinity of 1. In the following examples, the size of each dimension is 1 of an order of infinity of 1.

He then gives the example of a square, regardless of size, having an infinite number of such lines. The number of points is thus: ∞∞ However, another way of arriving at the square is taking an infinite number of points (line) times an infinite number of points (line), giving: ∞2.

The same goes for making a cube. Seen as an infinite number of squares made of an infinite number of lines, it is: ∞∞∞ However, the cube has several more representations: The simplest, ∞3, as well as ∞∞2.

If we now make two of these cubes, this can be represented in several ways, including 2∞3, ∞2∞2 and ∞∞2∞.

This is no longer the case when the value differs from 1, however; as far as the orders of infinity are concerned, it still applies, but equality will no longer hold regarding the finite quantities having these orders of infinity. It becomes clear that in the general case the correct representation is that of one side times the other - as with ordinary calculation - as opposed to the size of one dimension to the power of that of the other, though the latter could be done if suitable divisions were added to compensate the value.


Ouspensky writes of how the mathematics of finite and constant values cannot be applied to variable quantities, meaning cases where variables do not have a set value. However, if there are limited ranges of value, then it is of course quite possible to compute from these ranges the possible range of the result. But when values are also infinite, it does not work.

Working with orders of infinity, one can still establish definite relations between values that vary "infinitely", so long as they have a set order of infinity, or even a limited range of variable order of infinity, in which case one can arrive at the new range of variation in order of infinity of the result. One can also establish possibilities regarding the values of numbers that can vary "infinitely" in order of infinity, but that have set or range-bound values within whatever order of infinity they are of. Example of the latter:

x is 2 of any order of infinity; y is 3 of any order of infinity. The result of x + y will have either the value 2, the value 3, or the value 5, depending on which of x and y have the highest order of infinity or if they are the same, and the value will have that order of infinity.


That's all for now. Whether or not this is nonsense or meaningful remains to be seen. Feedback welcome.
 
You're dealing with some complex issues there.
I'll try to clarify the mathematical outlook on this, to the best of my abilities.

In mathematics, such liberal usage of infinity is not allowed.
One of the reasons is that it leads to false conclusions about infinity, some of which you came to.

This subject is mostly covered in the field of Mathematical Analysis which is a large and very complex part of
mathematics. There you will find a theoretical background for the questions about infinity (among other things).

I'll try to explain this problem briefly.

In maths, you can ask a question like "How many elements are there in a certain set?".
This number of elements is called the cardinal number, or sometimes the power of a set.

If we ask the question for natural numbers (1,2,3,...), the answer is, of course, infinity (this is designated "aleph zero").
The same question for real numbers, (natural numbers + fractions + irrational numbers + transcendent numbers) the answer is, again,
infinity (designated c or continuum).

But it has been proven that the power of natural numbers is actually less than the power of real numbers, or

aleph zero < c

So what does this mean? It means we have two infinites, one of which is "weaker" than another.
There is also a hypothesis in algebra that there are no other infinities between those two (or any higher than c).
Look for Cantor's theorem, and Dedekind Cut about this.

Okay, so that leads us to some of the conclusions you came to.

First, you implied that:

0/0 = 1
Inf/Inf = 1
(x*0)/0 = (x/Inf)*Inf = x
(x/0)*0 = (x*Inf)/Inf = x


In analysis it is not allowed to place infinity or division by zero in any expression like that.
Why? Because the end result depends on the expression (array/function/sum), or rather, the convergence of the expression!

For example you can have two arrays which both converge to infinity. But that doesn't mean that inf/inf = 1.
It can also be 2, 1/3, anything.
But this leads us to a subject of convergence (yet another part of analysis which deals with infinity).


Second, you state that (according to Ouspenski):

a) number of points on a line is infinte.
b) number of points in a square is infinity to the power of two.

Again, this is a typical pitfall which arises from dealing with infinities.
Total number of points on a line is infinite, that is okay. Actually, since line is continuous, the number of points equals c (continuum).
But, since there is no higher power of infinity than c, the total number of points in a square is, again, c, and not c-squared.
Yes, I know, it's mind-boggling, but it's true.

And not only that - if you take any line segment, like [0,10], then its power is c. But also, any of its subsets - like [0,1], also has c elements.
Or rather, the total number of points of any line segment is the same as the total number of points of an entire line.

I hope this helps, and I'm sorry for this lengthy post. I tried to keep it as brief as possible, but this is a complex topic.

EDIT: I wrote "proof" instead of hypothesis in one place...
 
Of the things you mentioned, I've read of one before in more detail, that being of the infinity of the number of elements of the set of natural numbers (or any infinite set of discrete numbers, it being possible to map elements in such to each other one by one going on infinitely, as it is possible to go on infinitely, thus making them equal in size) versus infinity of number of elements of real numbers or subsets thereof (where there are an infinite number of elements between any two elements, making mapping to an infinite set of discrete numbers impossible, as far as I remember the explanation).


Freelancer said:
There is also a hypothesis in algebra that there are no other infinities between those two (or any higher than c).
Look for Cantor's theorem, and Dedekind Cut about this.
Click to expand...

That I'll have to look up - though I have no idea how much I'll understand.

Freelancer said:
First, you implied that:

0/0 = 1
Inf/Inf = 1
(x*0)/0 = (x/Inf)*Inf = x
(x/0)*0 = (x*Inf)/Inf = x

In analysis it is not allowed to place infinity or division by zero in any expression like that.
Why? Because the end result depends on the expression (array/function/sum), or rather, the convergence of the expression!

For example you can have two arrays which both converge to infinity. But that doesn't mean that inf/inf = 1.
It can also be 2, 1/3, anything.
But this leads us to a subject of convergence (yet another part of analysis which deals with infinity).
Click to expand...

The way I was thinking was to define 'infinity by itself' (not part of any expression where it takes on a different meaning) as having a specific known value, and using the symbol to denote this. (and giving zero a corresponding inverse value)

Numbers were made to have in addition to the ordinary quantity a new quantity I called "order of infinity", those definitions I wrote for operations meant to 'work' when both these quantities are known.

With convergence, you have presently unknown quantities (if I understand things right) going somewhere, and where infinity represents a large but unknown value (not the specific, defined value which the infinity symbol represented in my thought) then the definitions I wrote do not apply. They were not really meant to.

What I'm wondering is mostly if those definitions would work in the context of being added to simple arithmetic, or otherwise used only where infinity (and perhaps best denoted by another symbol) means the specific value I made up.

Not that it would be terribly useful even if it did.

At least it was interesting to think about - hopefully wasn't wasting your and others' time by posting it - and nothing I'd read anyone describe in the same way (not that I'm anywhere near well-read mathematically, and most mathematical text being incomprehensible to me), though others have added new quantities to numbers to make them multidimensional in a large variety of ways.

Freelancer said:
Second, you state that (according to Ouspenski):

a) number of points on a line is infinte.
b) number of points in a square is infinity to the power of two.
Click to expand...

Actually, Ouspensky represented it as infinity to the power of infinity (infinite number of lines each having an infinite number of points), and then made this equal to infinity. I take the dubious "credit" for going "further" and adding into it the idea of infinity squared.

Freelancer said:
Total number of points on a line is infinite, that is okay. Actually, since line is continuous, the number of points equals c (continuum).
But, since there is no higher power of infinity than c, the total number of points in a square is, again, c, and not c-squared.
Yes, I know, it's mind-boggling, but it's true.
Click to expand...

It's beginning to intuitively make sense. There being an infinite number of points besides, infinitely close to, and between any points, there is no limit to possible points. And so, adding another dimension of the same does not change this, because there is already infinite possibility for more points.

Freelancer said:
I hope this helps, and I'm sorry for this lengthy post. I tried to keep it as brief as possible, but this is a complex topic.
Click to expand...

Given the subject, it seemed quite concise, and was well-worth reading. So, hardly any need to apologize.


And thanks for taking the time to set me straight.

EDIT: fix quotes
 
the idea that there are no higher infinities than 'c' is new to me. Is this a new result? I was under the impression there were a HUGE amount of infinities greater than 'c'.

wiki

http://en.wikipedia.org/wiki/List_of_large_cardinal_properties

Cheers
 
lamalamalamalama said:
the idea that there are no higher infinities than 'c' is new to me. Is this a new result? I was under the impression there were a HUGE amount of infinities greater than 'c'.

wiki

http://en.wikipedia.org/wiki/List_of_large_cardinal_properties

Cheers
Click to expand...

I was not referring to that in my explanation, but it seems to have produced a misunderstanding.
The total number of points on a line is continuum, c.
A square is made of lines, which are continuous. Actually, we can understand a square as a set of two dimensional points.
And how is the cardinal number of two multiplied sets defined?
As a Cartesian product of sets. |A|*|B| = |AxB|
But if we know that the cardinal number of A is a, and B is b, and if a,b are infinite, then
a*b = max(a,b)
But in case of a square, every set has a cardinal number of c, which gives max(c,c) = c.
What I meant is, there are no higher cardinals in that particular example, not "There are no higher cardinals in general".
Actually, it's not really known where c fits on a scale of cardinal numbers.

Those who are interested, can try looking up these "cardinal numbers" on the net, there has to be a lot of stuff written about it.
Although it's very interesting, it's also highly non-trivial.
I found this on wiki.

http://en.wikipedia.org/wiki/Cardinal_number
 
Great stuff. I was always confused when I was young, not knowing about specific rules to maintain order in mathematics when dealing with infinity.

When I was quite young I imagined infinity could exist between two points on a line or scale as on could continually halve (or factor with any other fraction) the distance forever. It really bothered me that infinity existed at the plus and minus extremes of the line and yet, there it was between two points.

I never was a good math student and yet it still fascinates me to this day. Sure wish I understood it better though.

Gonzo
 
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