Math and reality

[Exitwound]

This concept won't be new for some, but I still wanted to share:

How Gödel’s Proof Works

His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.

www.quantamagazine.org

But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.

It never ceased to amaze me, how mathematics correlates to psychedelic and spiritual revelations.
I think it is indeed, the true language of our reality.

 

[cubeananda]

Amazing stuff!

 

[downwardsfromzero]

It's wonderful - an incredibly precise tool that describes its own shortcomings by being unable to describe them.

 

[dragonrider]

I always wondered if the reason we tend to believe in the consistency of our own minds is because we're inconsistent.

 

[Jagube]

Thanks, in my logic course I didn't see a proof of Gödel's theorems, we only proved Turing's undecidability.

The article explains the proof of the first theorem well. In the second one it makes a bit of a leap, but I get it now. The missing bit is that if a set of axioms is consistent, you can prove in it that if it's consistent then G is true. And therefore you can't prove its consistency, because you could then prove G, which is not provable.

 

[Jagube]

This is a different approach to proving Gödel’s First Incompleteness Theorem, using Turing's computability theory:
A Computability Proof of Gödel’s First Incompleteness Theorem

Reductions are fascinating!

 

[dragonrider]

These problems of incompleteness or undecidability always seem to arise when systems reach a certain level of complexity.

I don't think you have these kind of problems yet, with something as simple as propositional logic.

If i'm correct, godel also wanted to prove that it can sometimes be possible in complex formal systems, to prove the truth of something that is actually false, but he just never got to that. It would have been the incorrectness theorem.

 

[Jagube]

These problems of incompleteness or undecidability always seem to arise when systems reach a certain level of complexity.

Well, natural numbers (aleph zero) are the precise level of complexity at which they start arising.

If i'm correct, godel also wanted to prove that it can sometimes be possible in complex formal systems, to prove the truth of something that is actually false, but he just never got to that. It would have been the incorrectness theorem.

If you can prove something that's false, then everything is true and that makes it an inconsistent system. So I don't know what could be there beyond that?

 

[dragonrider]

These problems of incompleteness or undecidability always seem to arise when systems reach a certain level of complexity.

Well, natural numbers (aleph zero) are the precise level of complexity at which they start arising.

If i'm correct, godel also wanted to prove that it can sometimes be possible in complex formal systems, to prove the truth of something that is actually false, but he just never got to that. It would have been the incorrectness theorem.

If you can prove something that's false, then everything is true and that makes it an inconsistent system. So I don't know what could be there beyond that?

I don't know. It was graham priest who said it, so i suppose it would have had something to do with paradoxes.
On the other hand, he may have just wanted to find a glitch in the system, a singularity.

 

[cubeananda]

I think I have found a neat way to demonstrate some of these principals.
This could be helpful to gain added perspective, that systems and their complexity are becoming less theoretical and more physical.

This visual loop.​

Spoiler

It can be described as a visually perfect loop, and also the shapes can be perfectly represented in purely numerical and arithmetical terms.

To date, complex formal systems can be extremely arbitrary and as such are able to prove things are are not true. To put it simply, the image above is the essence of the incorrectness theorem.

The only issue is at least one of the values in the systems has to numerically represent a human "Yes."

(Sort of like the Turing test with which most of us are now familiar, thanks to the movie Ex Machina )

In short, the image above appears to be a perfect loop, and it has been designed as such, but in reality the only way to generate the arithmetic required to produce that translation//rotation of points is by producing a much longer calculation. For the computer to generate this above image it would need the added rules of comprehending that the first and last frame are the same. For a human this is simpler. :roll: But perhaps this is because a human mind is in reality more complex of a system, only it deals in much larger and less binary values at a time?

 

[5 Dimensional Nick]

Fascinating stuff: i love it when math gets deep and trippy, thanks.

i have to leave this nugget of math here if u haven't heard of it, its one of my faves:

TREE(3), for a while a record holder of fastest growing function.

ok so huge number building functions and proofs use special symbols and special functions.

there is one called Knuth's up arrow.

put one arrow between two 3s and you have 16
put two arrows in and you have 65536
three arrows gives us a number with roughly 7 trillion digits in it!!!
i cant really easily describe the next one

TREE(x) grows faster though.

TREE(x) concerns trees that you can make from a certain number of colored seeds, the x in the function. you construct different trees and the number of possible unique trees gives you the output number (its actually a little more complicated but its too hard to explain for me)

TREE(1) is 1 (there is only one possible tree, the seed sitting alone)
TREE(2) is 3 (one of the seeds, the other one and the two connected to each other)
TREE(3) though is CRAZY. it IS finite but the number cannot fit in the observable universe even if every digit is written on a single particle. or even in a single planck volume. it probably cant fit in trillions and sextillions of universes but it is NOT infinite.

this blows my mind. but there's more:

using set theory you can prove that the proof in conventional maths that TREE(3) is finite, requires you to invent more functions (+, -, divide, multiply, e, squareroots etc.) than you can fit in the observable universe!!!!!!!!!!!!!

it just blows my mind that these conceptual structures are out there. things that dwarf the universe which is already so big it's hard to wrap ones head around.

and then there's always infinity!!!!!

 

[Jagube]

An excellent video that gives examples of self-reference, starting from Cantor's diagonal argument, and works up to the intuition behind the proof of Gödel's incompleteness theorems.
A delightful journey down the rabbit hole.

[YOUTUBE]