The simplest version of Godel's theorem and why it's important

In this video I will show you the simplest way to "get" Godel's theorem. Imagine an all-knowing computer (the limits of the thinking mind) that it can state any truth. Let's call it UTM for universal truth machine. Now if we write out G="UTM will never say G is true" and then ask UTM if G is true, we have put it to its limit. UTM cannot say it is true....which makes it true! The proves the limits of thinking and how there are truths beyond thinking.

[collijee]

Many thanks for this clear and simplified explanation which no doubt took much consideration and research. Yours was one of the first videos I came across when searching for material on Godel and I have come back to your video as it is indeed the simplest summary of this concept, not that any of the others were wrong, just that this is very succinct. Thanks again and the best to you!

[indio]

This is like a Benny Hill skit where Benny Hill m as a criminal, is brought before the Roman Emperor
The Emperor says to him,
"If you lie, I'll fee you to the sharks." If you tell the truth I'll feed you to the alligators."

Benny Hill replies,

"You'll feed me to the sharks."

[RalphHarnden]

Gödel proved that for any predicate logic system that includes the equivalence relation and the successor function (I.e. simple arithmetic), and whose axioms are consistent, one can construct a well formed formula that asserts its truth value… that if it’s true, then it is false, and vice versa.
In other words, consistency implies incompleteness.
Rosser later proved that completeness implies inconsistency, showing that completeness is equivalent to inconsistency.
See Kleene: Introduction to Metamathematics.

[RalphHarnden]

I also find Lorenzen’s intuitionistic logic approach to proving the Fundamental Theorem of Calculus very interesting.
He does not derive a contradiction to prove it. Instead, he uses a Cauchy sequence.

[leonelmateus]

Penrose explained it beautifully. My take from him is that a system of proofs cannot itself by proven using that same system. hence making it incomplete. i.e. It requires an external truth verifier or a system or knowledge or meta-knowledge outside of the system of proof being utilized. you don't really explain this, you demonstrated that you can make the utm fail logically. it is also not a point on recursion, this is also a point of confusion (often conflated).

[Evergreen-sllj]

I suspect I missed something, but isn’t that like Pinocchio saying “My nose will grow”?

[asuited]

Correct me if I’m wrong but wouldn’t the observation of UTM not responding to the question be the proof itself of the statement being true? That in itself is still UTM answering the question correctly

[nathangonzales]

Need a PhD to understand.. Typical gatekeeping of ideas through overcomplicated codification. Mathematicians seem to be the worst offenders, simultaneously monopolizing and obfuscating Gödel's ideas.
Simply put:
All descriptions of reality are inadequate, and any description of reality can, at best, be internally consistent.

Knowledge can then only be either:
- Indirect observation ( consequence of a series of events of which unknowns will always be present )
- Definition (descriptors - language, ideas, math, logic, models, categorization)

Gödel's incompleteness theorems are limits for knowledge and language.

[-_Nuke_-]

To explain what's being said in the video:

Assume a "G" statement about the Universe. Is it "true" or is it "false"?
We have the "UTM" machine that knows everything...

Lets also invent a proposition A. Where A = "UTM will never say that the "G" proposition is true"
We will now ask UTM to tell us if A is true or false.

This goes all well and good... UTM will look at G, determine if its true or false and tell us...
But what happens if we then name the statement ""G" instead of "A".

So we ask UTM again

We say to it... Take G = "UTM will never say that the "G" proposition is true"

And ask UTM...

"Is "G" true or false?"

The implication that I can see from this, is that mathematics is also ultimately a language. A very sophisticated one, but a language still. Therefore it will also fall into the linguistic paradoxes like

A: This statement (A) is false.

If (A) is true, then "This statement is false" is true. Therefore, (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then "This statement is false" is false. Therefore, (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.

Now... Even thought we know that these paradoxes exist, it doesn't hinter us at all from using English or whatever other language to achieve the civilization that we have achieved as Human species... Is English incomplete because of that?

The "A: This statement (A) is false." statement, can't be proven inside English or any other linguistic sort of model... But its also a statement of zero value... Yes, we can't prove that from "bellow" but we don't care.

I argue that there is not a *single* mathematical theorem of even the slightest significance that we will not be able to approach because of Godel's incompleteness... Ok... Yes, Godel, you win... We can't prove gibberish using mathematics, but we also don't really need to...

[Jupa]

I think Godel's Incompleteness theorem is the key to understanding the universe
Thank you for the explanation, there are many but this highlights why it's so significant and marries the field of mathematics/logic with philosophy/human condition.

[chris-sixty]

I made an audio file of myself repeating positive affirmations (ie. everything always works out etc) so as I'm listening to myself on headphones, my uplifting words really sound/feel like my actual thoughts. And then it hits me - what if our thinking mind replayed positive statements like this over and over all day long? without even having to try? Ie. Today will be amazing, there is nothing to fear etc etc But the thinking mind does just the opposite. So the eureka moment for me was - if I COULD choose my thoughts, I would choose exactly what I had said in this affirmation audio file I had made. And because my "thinking mind" sounds nothing like that - it was more proof that my thinking mind is not the REAL me

[nameless-ydko]

The First Law of Soul Dynamics is;
"For every Perspective, there is an equal and opposite Perspective!"
All that exists is perceived.
There is nothing to perceive other than Truth/Mind/Concepts/Memory!
The One Mind is perceived by the One Consciousness via all of us unique Perspectives (Souls).
The only thing that is 'incomplete' is that which is perceived at any one moment by any individual unique Perspective! ;)

[jamestagge]

If one gives UTM the statement, “G=UTM will never say that G is true” and asks if it is true, the machine then confronts a paradox akin to Quine’s liar paradox, “this statement is false”. But this is sophistry and not logic. It is akin to saying that “I think I am not thinking” which is a statement that, again, appeals to truths to formulate a position which denies the existence of truth. It defeats the means of its own definition/formulation. When this supposed UTM is given the statement G (“UTM will never say that G is true”) and then asked if G is true, what has happened? If the UTM were actually a machine or some manner of computer, it could not be given such a data string, if that is the proper terminology, G=the UTM would never say G is true.” Computers do not function this way. There is nothing to judge in such a statement, no calculation which would be either true or false to determine by which to then output an answer. For example, one would have to input in to some manner of program of analysis, G=”false”. A query then in some manner of computer operations which would request information about G would yield, G=”false” and there would be no paradox/self-reference. The UTM wouldn’t “say” anything if a machine. It would require data to analyze to arrive at a value to output.

Or, if we consider that the UTM were some manner of magical system which would respond not to data as in the above, but an expression such as the semantic, “G= “the UTM will never say that G is true”, there is nothing about this statement which can be analyzed to determine if it is true or false, but only that it represents an assumption that the UTM will not be able to respond that it is true notwithstanding that there is no truth value to the statement at all, i.e., it is neither true nor false, so why would the UTM never say that G is true? Is it because it is false? Is it because it is true? It has to be one or the other or the UTM has been given a programmed response which is based on nothing, i.e., no truth or falsity to be or that has been determined. Again, “I think I am not thinking”.

I am sorry but this is piffle and to be pointed about it, sophomoric. It makes no sense whatsoever.

[unexpectedTrajectory]

Not just trying to be argumentative, but I don't think this is Godel's theorem. Not really even kind of. What you just stated is just a Linguistic paradox that includes a confounded oracle machine. The critical thing about Godel's theorem is that is has to do with provability, not truth. So you can see that the statement IS true (not paradoxical), but can never be proven (thus the incompleteness - there are true statements which cannot be proven.)

Godel's statement was a mathematically precise version of "this statement cannot be proven." Now if you CAN prove it, then you have a contradiction, and the system is inconsistent. But it's fine for it to just be true. A true statement that cannot be proven, which proves that any system capable of the statement is necessarily incomplete.

I think your example is actually a demonstration of the set theory paradox ("the set of all sets that don't contain themselves" doesn't contain itself iff it does, and does contain itself iff it doesnt), because your example could be recast "this statement is in the set of statements UTM will affirm are false" is false if it's true and true if it's false.

Thoughts?

[VioletDeliriums]

you should play that evh guitar only in situations where ironic distance is the desired effect.

[enlilannunaki]

Somehow he linked a simple logic paradox to mathematical proof; that is where my simple mind gets lost.
The statement is a mathematical axiom that is intrinsically true, but cannot be proven.

[jeffbguarino]

I don't understand the problem. You can have things that are true and false at the same time. As in Schrodinger's cat who is dead and alive at the same time. It seems like the mathematicians are confused about things and creating a paradox but these things are common everyday in QM. In the town where the barber shaves all those who do not shave themselves, who shaves the barber? This is also easy, just put the barber in an isolated box with an electron emitter and spin detector. Tell the barber to shave himself if the spin is up and not to shave himself if the spin is down. When you close the box the barber is in a superposition of having shaved himself and not shaved himself at the same time. The set R that is the set of all sets that don't contain themselves. Write set R1 on a piece of paper, R1 is the set of all sets that don't contain themselves and include R1 in the set. R2 on a 2nd paper and R2 is the set of all sets that don't contain themselves and don't include R2. Put both papers in a box and have an electron spin detector which when spin up will burn the first paper and spin down will burn the 2nd paper and you will have a set in this box that is a superposition R3 and it will have R3 and not have R3 in the set at the same time.

In the double slit experiment the electron goes through the left slit and it does not go through the left slit at the same time. This is why everyone says QM is weird.

So why is it mathematicians can't have things that are true and false at the same time ? It is an everyday occurrence in the real world. They might think it is weird to have something true and false at the same time. Is this the reason that they don't want to have math statements that are true and false at the same time ? Because it is weird ? Welcome to the real world is what I say.

[Jalcolm]

This is why introspection is impossible. The mind is for seeing, but not itself. When it attempts to see itself, time passes and it sees the memory of itself. I think, therefore I was. Time is a mirror.

[GeekOverdose]

Isnt it like a version of the liars paradox? A self referential setup creates a paradox

Perhaps it should not be called a paradox. Maybe its not a proposition to begin with. Since if you unwrap it, its just

"UTM will never say "UTM will never say "UTM will never say "UTM will never say "UTM will never say is true" is true" is true" is true" is true" so I would expect it to just freeze as it is processing these infinate recursions

[barouchkrakauer]

Many many many years ago, Jews asked: "Can God create a stone he cannot lift?"