Kurt Godel - World's Most Incredible Mind

[Pigeon]

Kurt Godel: The World's Most Incredible Mind.

"Either mathematics is too big for the human mind or the human mind is more than a machine" ~ Godel

1. If the system is consistent, it cannot be complete.
2. The consistency of the axioms cannot be proven within the system.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system.

If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true.

Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system.

To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as Gödel numbering.

THE first video gives a background on math theory and persons leading up to Godel.





Thread on his Incompleteness Theorem here

[Pigeon]

[Pigeon]

Does that Incompleteness Theorem point to this being a simulation?

[Pigeon]

So there is always something outside the system.

[Royal]

Amazing stuff... thank you.


I found the following the most fascinating:

From the first set of videos:

Infinite sets if Infinity
I was thinking about this for the past couple months regarding personal, and individual, "timelines" of everyone on the planet. That means on some other timeline, I am a employed computer scientist instead of an unemployed accountant. The Syria crisis never happened, or the theorized "meteor" never wiped out the dinosaurs.

Higher sets of Infinity
As 1.1 , 1.2., 1.3 will give a denser set of infinity than 1, 2, 3, we can see the advantages or disadvantages of the sets of infinity.

1.1, 1.2, 1.3
More Detailed
Longer
Slower

1, 2, 3
Less Detailed
Shorter
Faster

Because of this fact of measurement, I disagree with whoever calls it "higher set of infinity" as its definition. I would just say "different" sets of infinity. We can all remember a time, when we wanted a moment in time to be quicker or slower.


Blind spots will exist if consistent
I thought this was an interesting duality as it applies to people. How often are the most consistent people- those that follow a prescribed path of school and work, are sometimes the least conscious of the depth of the world around them. There is a saying "Those that grow up in the best neighborhoods, and go to the best schools, and get the best jobs are people who will know reality the least". The same can be said in the opposite direction "worst neighborhoods, worst school,... etc..."

"Coming back to yourself at an earlier moment in the timeline"
I meditated on this thought before, regarding why is it that some people can get a sense of the future. If we split from a higher source, this would mean that we also combine again to form higher beings- according to Godels theory of a circular timeline. If we achieve a godlike nature to control time and space, we would have to use conscious gateways to interact with higher selves.





Second set of videos:

There is always something outside the system
We have discussed such topics in our skype chats. Is the human race a cover for a more advanced civilization? Do we have the capability to prove it? Can mankind invent technology to overlap reality with another reality, and are we in one now?


Incompleteness thereom has implications for Religion
Mix this in with infinity of infinities and religion can become true (I think). But that would also make it false! HAHAHAHA... I love this incompleteness stuff! And religion cannot prove if those forces outside themselves are good or evil.

The debate would be, there are Gods (obviously an infinite amount), but what are their intentions? I guess that would be up to us, because we are creating them and empowering them, calling them down, kicking them out, becoming... there the janitors, teachers, CEO's, firefighters, scientists, physicist, counselors, accountants, lawyers, unemployed, homeless, wives, daughters, husbands, sons, family, associates, colleges, and disciples...

And they are all human.

[Royal]

Does that Incompleteness Theorem point to this being a simulation?

A dangerous concept if not taught properly. People can use this theory to support their selfish interests. To say that it's so limited that it can't interact with other "live" simulations would be foolish. Or that it would NOT be able to "flux" move in and out of reality and simulation would be foolish.

I propose my theory of "the real becomes fake, and the fake becomes real" to analyze the simulation situation. For example, how real it was to get the right shoes for every year of high school. But how fake it is now? I can wear a good pair of shoes three years straight. Obviously, a hierarchy of value --> what is real value and what is fake. So what attitudes or patterns of thinking are actually fabricated from a higher reality? The reality being, we may need something on our feet to keep our feet from being cut.

Or for a more accurate example of the concept: If I present a created human from advanced 3-D printing technology instead of it coming out of a womb, that is indistinguishable in physical and non-physical properties (thoughts) from a real human, how real would it be? Although fabricated, everyone treats it as real, and it grows to have real feelings, thoughts, and a life of it's own and procreates. Nobody would disagree with the realness of it.

So the process went: Real Human --> Created Fake HUman --> Seen as Real Human

And the process would then be:

Real Universe --> Fake Universe --> As real as any universe with real consequences

[Pigeon]

Start with this example.

World's shortest explanation of Gödel's theorem

From Smullyan, is the World's shortest explanation of Gödel's theorem.

We have some sort of machine that prints out statements in some sort of language. It needn't be a statement-printing machine exactly; it could be some sort of technique for taking statements and deciding if they are true. But let's think of it as a machine that prints out statements.

In particular, some of the statements that the machine might (or might not) print look like these:

• P*x (which means that the machine will print x)
• NP*x (which means that the machine will never print x)
• PR*x (which means that the machine will print xx)
• NPR*x (which means that the machine will never print xx)

For example, NPR*FOO means that the machine will never print FOOFOO. NP*FOOFOO means the same thing. So far, so good.

Now, let's consider the statement NPR*NPR*. This statement asserts that the machine will never print NPR*NPR*.

Either the machine prints NPR*NPR*, or it never prints NPR*NPR*.

If the machine prints NPR*NPR*, it has printed a false statement. But if the machine never prints NPR*NPR*, then NPR*NPR* is a true statement that the machine never prints.

So either the machine sometimes prints false statements, or there are true statements that it never prints.

So any machine that prints only true statements must fail to print some true statements.

Or conversely, any machine that prints every possible true statement must print some false statements too.

The proof of Gödel's theorem shows that there are statements of pure arithmetic that essentially express NPR*NPR*; the trick is to find some way to express NPR*NPR* as a statement about arithmetic, and most of the technical details (and cleverness!) of Gödel's theorem are concerned with this trick. But once the trick is done, the argument can be applied to any machine or other method for producing statements about arithmetic.

The conclusion then translates directly: any machine or method that produces statements about arithmetic either sometimes produces false statements, or else there are true statements about arithmetic that it never produces.

Because if it produces something like NPR*NPR* then it is wrong, but if it fails to produce NPR*NPR*, then that is a true statement that it has failed to produce.

So any machine or other method that produces only true statements about arithmetic must fail to produce some true statements.

Does this mean that there are truths we will not know or are some of these truths actually wrong?

In the video where the man speaks of things outside the system, I don't know if he working from:

-- That there are truths we will not know inside the system, but outside.

in other words

-- To understand the system we must know something from truth outside the system (understand NPR*NPR*)

Does this lead to 'god', don't know.