M2C2A: Episode 13, ∞, what has always been wrong about it

Many say ∞+1, ∞², or ∞-10^(10^(10^(10^10))), are simply ∞, but no.

∞ is like i, you (usually) can't simplify things with it. E.g.
If ∞+∞ is ∞, then we would be obliged to say ∞-∞=∞.

Is it?

We can use something I like to abbreviate as P.I.G (Patterns In Graphs, which can also refer to look for formulas), which is what I'll do.

If we graph x-x=y, we always get 0.

It's a truly linear equation.

We don't expect that when it gets to ∞ it will suddenly rise to ∞ too.
Same story with ∞*∞ and ∞/∞.

What makes sense in both cases, is that neither ∞+∞ or ∞*∞ should be expressed as ∞, but as 2∞, and ∞², respectively.

So ∞+1 should be written as it's written here, and so should 3/∞, 2^∞+1, x^y=y^∞x, or ∞x²+∞x+∞=0, etc.

...or is it?

This would work perfectly if it wasn't for the definition of ∞, the biggest quantity that can be described, which doesn't have an end.
If ∞+1>∞, then ∞+1 should be infinity!

But if ∞ (∞+1)<∞+1 (∞+2), then ∞+2 should be ∞!
But if this is smaller than ∞^∞(totally infinite!), then ∞^∞ SHOULD BE ∞!!!
So my definition of ∞ is:

nonsense.