Let's talk about Infinity.

∅ -> Nothing
ξ -> Everything

The Empty Set implies Nothing, while Everything implies, well Everything. Please note that I did not use the standard symbols for Zero or Infinity, an equal sign, or anything approaching Mathematical Rigidity.

Also, these definitions won't work, as the above sort of implies (to me, anyway) that it might be possible to get from Nothing to Everything by counting or something. But that won't work, as that's not the type of Nothing and Everything that I am talking about.

So, let me redefine my terms.

∅ -> Absolutely Nothing
∅ -> An Imponderable Sort of Nothing
ξ -> Absolutely Everything
ξ -> An Imponderable Sort of Everything

And when I say an Absolute Imponderable, I mean an Absolute Imponderable.

∅ + 1 = Nonsense
What is the One being Added to?
ξ + 1 = Nonsense
Where is the One coming from?

Or to state that differently:

∅ doesn't really exist.
ξ is all that exists.

And none of the Standard Operators (+, -, *, /, =) work on either.

In fact, only one Operator works on Nothing and Everything: The Not Operator.

Nothing and Everything are the Compliments of one another. To talk about The Void precludes Existence. And to talk about Everything else (or even just the smallest part of The Whole) precludes The Void.

Or perhaps, in better English, if one takes away The Void, one is left with Everything Else. And if The Totality of All is Negated, one is left with Nothing.

I question whether these (∅ & ξ) are useful concepts. About the only other statement of any interest (and I question whether this is of interest) that my mind can (currently) formulate is:

Meaning, at this level of inquiry, I would be tempted to leave either ∅ or ξ behind and simply reduce the One to the Negation of the Other and leave it at that.

Having introduced two Imponderables that are almost completely useless, it makes a perfect sort of sense to now use them in a Mathematical Equation to Define our good friends Zero and Infinity.

These are Definitions!

Both 0 and ∞ derive directly from our Base Imponderables (∅ & ξ). And thus, they are Equally Imponderable. So, don't worry your pretty little head about them... I mean, unless you want to.

Do you want to?

Fair enough. That's why you are here. With both 0 and ∞ we can do some Math.

### Operations On Zero

Let's start with Zero.

Zero acts as the Identity Function for both Addition and Subtraction.

Hey, ma! Look at me! I'm using fancy Mathematical Terms.

And everything else (except for Infinity, as I'm not so sure about Infinity) acts as a Multiplication Identity Function for Zero.

Hey, we did +, -, & *. And we have used =, so I wonder what would happen if we divided by 0?

What do you know? The world did not explode. Also, there are plenty who would disagree with me. So, let me present my argument.

Division is a Per Container Operator.

You have Two Bananas.
I have Three Bananas.
I am a Capitalist.
I now have Five Bananas.
And you have none.

Oh, did that not provide clarity? Then please, let me continue... using fancier nomenclature and needlessly foreign subscripts, because we are Mathematicians, are we not?

3_{βme} + 2_{βyou} = 5_{βme} + 0_{βyou}

Typically, folks just drop the Trailing Zero. But it's important to realize it is there. In Real World Addition, there are always Types.

I have One Banana.
I'm a Great Guy.
My Friends have None.
So, I share with them equally.

Let's look at that in the fancier nomenclature I've decided to use.

1_{βme} + 0_{βx} 0_{βy} = ⅓_{βme} + ⅓_{βx} ⅓_{βy}

Clearly, if enough folks share a single Banana, the share of a Banana each will get falls to Zero.

Conversely, as the number of folks sharing a single Banana falls to Zero, the amount of Banana each of these fractional people gets rises to Infinity.

1_{βme/10} = 10_{βme/1}
1_{βme/100} = 100_{βme/1}
1_{βme/1000} = 1000_{βme/1}
1_{βme/∞} = ∞_{βme/1}

Essentially, I am allowing Division between 0 and ∞ to be a two way Identity.

Yes. Yes. It doesn't work. And it may well won't.

But I have a hunch (call it an intuition, all the Math Geeks are doing it) that if Types are Strictly Enforced, no Paradox or Impossible Outcome will result.

Furthermore, I've never encountered a Computational Problem wherein Dividing by Zero did not result in Infinity, if (and this is a big if) the result was utilized as a final value: i.e. something that was never used for further computation.

### Operations On Infinity

As before, let's start with the simple stuff.

Addition and Subtraction on Infinity are Identity Functions.

The argument for this really gets down to a problem of definitions.

If ∞ is a number,
then ∞ + x = ∞ + x

But rather than a Number, I view Infinity more as a concept... lying somewhere on the plane of ideas between an Integer and Everything.

Which means, I'm sort of OK leaving Addition & Subtraction on Infinity hanging.

So, let's just move on to to division.

Once again, there really is no proof of this. But the following subsets of The Positive Integers are generally considered to be Equally Infinite in Length.

Integers
Positive Integers
Negative Integers
Odd Numbers
Even Numbers
Prime Numbers
Squares
Cubes

In fact, I am sure there are Infinite subsets of Integers that are all Equally Infinite in length.

Thus:

And here, it may make some sense to recap.

∑^{∞}0 = ∅
∑^{∞}∞ = ξ
0 + x = x
0 - x = -x
0 * x = 0
x / 0 = ∞
∞ + x = ∞
∞ - x = ∞
∞ * x = ∞
x / ∞ = 0

And the only thing left to decide is Zero Multiplied by Infinity, which I am happy to leave as undefined, as it's essentially a Nonsensical Question, being a Type Mismatch... or so I will nonchalantly state.

0_{x} * ∞_{y} = 0_{x} ∪ ∞_{y}

Finally, let me revisit our opening definitions, adding a Types Nomenclature.

∑^{∞}0_{i} = ∅
∑^{∞}∞_{i} = ξ

Meaning, The Void is equivalent to Every Type (and/or All Combination of Types) having a Value of Zero.

While, The Entirety of All is equivalent to Every Type (and/or All Combination of Types) having a Value of Infinity.

∅ is The Nothing Beyond Nothing
ξ is The Totality of Everything

These are, of course, but Definitions.

Upon further consideration, I question whether I actually believe anything much beyond:

∑^{∞}0_{i} = ∅
∑^{∞}∞_{i} = ξ

I probably should have stopped there.

And in Truth (if we are looking for Truth, Bedrock Truth), my definition of The Void seems overly inclusive. And I might just want to redefine my terms as:

∑^{0}0_{i} = ∅
∑^{∞}∞_{i} = ξ

I certainly would not include any Mathematical Operations on Zero or Infinity at this juncture, as they so obviously lead to Computational Paradoxes.

But all that said, I still like the base idea of Types on Infinity and think there is some underlying Mathematical Truth in The Conception.