[hamdani yusuf (OP)]

H3 is exponentiation. We can expand rational numbers to some part of complex numbers by adding root numbers. Here's an example.
x ^ 3 = 2
In Hyperoperation notation,
x (H3)  3 = 2

There are 3 solutions for x.
x=2^(1/3) ≈ 1.2599
x=(-1 - i√3)/(2^(2/3)) ≈ -0.62996 - 1.09112i
x=(-1 + i√3)/(2^(2/3)) ≈ 0.62996 + 1.09112i

None of them are rational number.
What we usually call algebraic numbers are finite additions of one or more root numbers which have 0 imaginary component, or 0 argument in polar coordinate.
For example, 1+√2 cannot be expressed as a simple root number, which can be turned to an integer by raising it to the power of a rational number.

[hamdani yusuf (OP)]

Going from H3 to H4 has a problem, because exponentiation is generally non-associative. It's shown in the examples below.

plot x^(x^x) - 2
https://www.wolframalpha.com/input?i=plot+x%5E%28x%5Ex%29+-+2

plot( x^x)^x - 2
https://www.wolframalpha.com/input?i=plot%28+x%5Ex%29%5Ex+-+2

If the brackets are omitted, Wolframalpha seems to evaluate the equation from the right most exponentiation first.

[hamdani yusuf (OP)]

A totally different way to do math

There's a strange number system, featured in the work of a dozen Fields Medalists, that helps solve problems that are intractable with real numbers.

I'm wondering if this number system can help solving the problem here called "the continuum hypothesis conundrum".

[hamdani yusuf (OP)]

The videos below can help us understand about p-adic numbers.

How to Wrangle Infinity (an intro to p-adic numbers)

Intuition for the p-adic metric

[Zer0]

If you don't mind, i have a small query.

As the Topic of the thread is 0...

I was Wondering..

Are there any fields of knowledge in which 0 is attributed a (+) or (-) sign?

Have you ever come across a +0 or -0 value?

Thanks!

[hamdani yusuf (OP)]

If you don't mind, i have a small query.

As the Topic of the thread is 0...

I was Wondering..

Are there any fields of knowledge in which 0 is attributed a (+) or (-) sign?

Have you ever come across a +0 or -0 value?

Thanks!

I've seen them used in the study of limits.

[hamdani yusuf (OP)]

What we usually call algebraic numbers are finite additions of one or more root numbers which have 0 imaginary component, or 0 argument in polar coordinate.

The argument in  polar coordinate can also be π to make the number real.

[hamdani yusuf (OP)]

Math Has a Fatal Flaw

Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer.

The video title sounds bombastic, perhaps deliberately to increase views.

IMO, the paradox and confusion comes from how we treat infinity.

https://brilliant.org/wiki/infinity/
Infinity is the concept of an object that is larger than any number. When used in the context "...infinitely small," it can also describe an object that is smaller than any number. It is important to take special note that infinity is not a number; rather, it exists only as an abstract concept. Attempting to treat infinity as a number, without special care, can lead to a number of paradoxes.

Infinity is not a number!

Because infinity is not a number, it requires special rules of arithmetic in order to remain consistent. This is a useful view in fields such as measure theory, which extends the real numbers by adding two numbers ∞ and −∞, that satisfy some additional rules:

a+∞ = ∞+a = ∞ (for any a besides −∞)
a−∞ = −∞+a = −∞ (for any a besides ∞)
a⋅∞ = ∞⋅a = −∞ (for positive a)
a⋅∞ = ∞⋅a = −∞ (for negative a)
a/∞ = a/-∞ = 0 (for real a)
∞/a = ∞ (for positive a)
∞/a = -∞ (for negative a)

It is worth noting that 1/0 is not ∞. Additionally, operations involving multiple infinities (such as ∞−∞ and ∞/∞) are not generally well-defined.
Importantly, note that this is an extension of the real numbers chosen specifically to allow infinity to be treated as a number; without this context, infinity remains a concept rather than a number. For instance, the following sections continue to treat infinity as a concept rather than a number, because the measure theory context above no longer applies

Accepting the validity of diagonalization method comes with the cost of flawed and broken math. The diagonalization method contains loop holes like what we find in "proof" demonstrating that 1=2.

[hamdani yusuf (OP)]

"Prove" 2 = 1 Using Calculus Derivatives - Can You Spot The Mistake?

[hamdani yusuf (OP)]

Any number can be represented with a finite amount of information or digital bits of information.   It's just that you can't have this ability for all numbers simultaneously.

I don't think it's true. If a number has infinitely long digits with no algorithm to express those digits, it cannot be represented with a finite amount of bits. Algorithmic numbers can be represented with a finite amount of information. They are essentially compressible numbers. On the other hand, non-algorithmic numbers are incompressible. There are infinitely many more incompressible numbers than compressible numbers.

[Origin]

Have you found an answer to the question "how many numbers exist"?

[paul cotter]

"As many as you want" or words to that effect is what I suggested long long time ago.

[hamdani yusuf (OP)]

Have you found an answer to the question "how many numbers exist"?

Yes, it's infinitely many.
Infinity itself is not a number. It's an abstract concept. Treating it as a number like what Cantor did led to contradictions.