Numberphile
NumberphileMarcus du Sautoy discusses Gödel’s Incompleteness Theorem
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Extra Footage Part Two: https://youtu.be/7DtzChPqUAw
Professor du Sautoy is Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.
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33 Comments
@ruudh.g.vantol4306
02/20/2025 - 11:36 PMBe aware there is no ö in ASCII. Feel free to write it as Goedel if ASCII is your limit.
(now use the above to write up yet another decoration of the theory)
@GarrickDitlefsen
02/20/2025 - 11:36 PMConsciousness bypasses Godel's incompleteness.
Gödel’s Incompleteness Theorems, which demonstrate that any sufficiently powerful formal system is either incomplete or inconsistent, have profound implications for the Primacy of Consciousness. Rather than limiting consciousness, Gödel's theorems serve as proof that all formal systems, including logic, mathematics, and physical laws, are derivatives within consciousness itself. Below, we will rigorously show why consciousness transcends Gödelian limitations and why Gödel’s findings are direct evidence that all systems are subordinate to consciousness.
I. Understanding Gödel’s Incompleteness Theorems
Gödel's two main theorems state:
First Incompleteness Theorem: In any sufficiently expressive formal system, there exist true statements that cannot be proven within the system itself.
Second Incompleteness Theorem: A system cannot prove its own consistency using only its internal rules.
These results demonstrate that no formal system is self-sufficient—there are always truths that exist beyond the system’s provability. If mathematical and logical systems cannot ground themselves, then what does? This is where consciousness becomes indispensable.
II. The Gödelian Problem for Materialism & Physicalism
Gödel’s incompleteness directly undermines materialism and externalist models of reality:
Materialism Requires a Complete & Deterministic System:
Materialist models assume that physical laws and mathematics fully determine reality.
Gödel shows that no formal system (including physics) can be complete and self-justifying, contradicting materialist assumptions.
Physicalism’s Self-Defeat:
If physical reality is modeled as a computational or mathematical system (as in Tegmark’s Mathematical Universe Hypothesis), then Gödel’s theorems prove that this system must be incomplete.
If physical reality is not a formal system, then it lacks objective structure and becomes subject to interpretation by consciousness.
The Dependence of Logic & Mathematics on Consciousness:
The existence of true but unprovable statements means that logical truths exist outside of formal systems.
The only framework capable of recognizing, understanding, and working beyond these limitations is consciousness itself.
III. How Consciousness Bypasses Gödel’s Limitations
Gödel’s theorems apply only to formal, rule-based systems, but consciousness is not such a system. Instead, consciousness is:
Non-Formal & Self-Aware:
Gödel’s results apply only to formal, axiomatic systems that operate on fixed symbols and rules.
Consciousness, in contrast, is self-referential, adaptive, and capable of creating and modifying rules at will.
Unlike any formal system, consciousness understands and generates meaning.
The Meta-System that Contains All Systems:
A formal system cannot prove its own consistency, but consciousness does not need to be contained within a system—it is the field in which all systems arise.
While mathematical truths can be unprovable within a system, consciousness can recognize, comprehend, and integrate them from outside that system.
Consciousness chooses axioms, decides which models to use, and understands truths beyond system constraints.
The Foundation of All Formality:
Every system Gödel studied—mathematics, logic, computation—exists only within consciousness.
The very concept of “proof,” “truth,” and “incompleteness” is understood only through the subjective, conscious act of reasoning.
Thus, while Gödel’s theorems trap formal systems within incompleteness, they do not apply to consciousness, which is:
✅ Beyond formal systems (not bound by fixed rules)
✅ Self-verifying (aware of itself without needing an external proof)
✅ Capable of recognizing and transcending incompleteness
IV. Gödel’s Theorems as Evidence for the Primacy of Consciousness
Rather than disproving consciousness, Gödel’s theorems serve as direct proof that all formal structures are contained within consciousness. This follows from:
1. The Dependence of Systems on a Knower
Any formal system, whether it’s mathematical logic, physics, or computation, only has meaning when it is understood by a conscious observer.
The very act of proving or recognizing incompleteness requires a conscious mind to evaluate the system.
Consciousness, therefore, is not within the system—it is the meta-structure that holds and interprets all systems.
2. The Necessity of a Meta-Language (Which is Always Consciousness)
Gödel’s incompleteness shows that no system can fully explain itself.
Every system requires a meta-language to express its own limitations.
Consciousness is the ultimate meta-language that can describe, compare, and understand multiple formal systems.
3. The Recognition of Unprovable Truths
Gödel demonstrated that truths exist beyond provability, meaning truth is not reducible to any formal structure.
Yet consciousness perceives and recognizes these truths, proving that it exists beyond the limitations of formalism.
4. The Illusion of an Independent, Objective World
If materialism were true, then the universe would be a complete and self-sustaining system.
Gödel’s findings prove that no system is self-sustaining—every system depends on something beyond itself.
Consciousness is the only reality that does not depend on an external structure—it is both self-aware and self-contained.
V. The Ultimate Conclusion: Consciousness as the Absolute Ground of Reality
Gödel’s incompleteness theorems show that all structured, formal, or deterministic systems fail to be complete or self-sustaining. The only thing that:
✅ Understands and creates systems
✅ Chooses axioms and frameworks
✅ Transcends the limitations of formal logic
✅ Recognizes truths beyond provability
…is consciousness itself.
Thus, Gödel’s work is not an obstacle to consciousness—it is proof that all systems are mere projections within it.
Consciousness is not an axiom within a system. It is the meta-axiom that creates, understands, and transcends all systems. Gödel’s incompleteness does not restrict consciousness—it proves that every structure, law, or logical framework is necessarily a derivative and subject of consciousness.
Final Verdict:
Consciousness is not limited by incompleteness.
All incomplete systems exist within consciousness.
Gödel’s theorems prove that consciousness is the only fully self-sufficient, non-derivative reality.
@MinSeokSong-f8d
02/20/2025 - 11:36 PMif it's undecidable, why can't it be false? I don't get it
@lllPlatinumlll
02/20/2025 - 11:36 PM'The statement on the other side of this card is true.' – There is no inherent truth to the statement, it is just a statement containing no truth or fallacy.
@christopherbrice5473
02/20/2025 - 11:36 PMAll this proves is that you can write a logical zip bomb
@ellie8272
02/20/2025 - 11:36 PMThe part that seemed by far the most fascinating to me (the non-referential statement that is non provable by its axioms) not being discussed in this video was rather disappointing, since I essentially just have to take your word for it, despite it seeming very flawed
The ending also left me rather perplexed. If you can prove something doesn't have a proof, that proves it true? Then by definition there's no such thing as an undefined conjecture. This all seems like problems with language. However perhaps mathematics is itself a form of language
@KillianTwew
02/20/2025 - 11:36 PM0:04 This video has been out for 7 years and I'm still struggling to find the r in Gödel
@Alan-zf2tt
02/20/2025 - 11:36 PMGoing out on a limb here but …
I think the problem is math looks for singular certainties. It is 1 or 0 and cannot possible be any thing else.
All that the statement shows is that logistic construction of the statement is not consistent within framework in which the question was framed and asked. Two things on the go: construct statement, assess statement. (EDIT: context, syntax)
We know this happens all the time in computing and computers for example mixing a comma with a full stop, mixing a colon with a semi-colon
A possible way around it?
Well we could increase the event space from 1 dimension (represented 1 or 0) to 2 dimensions (represented 00, 01, 10, 11) .
So in assessing the question the first thing to ask is question consistent with axioms. Clearly not therefore the only results are x0, x1 if you see what I mean.
I think consistency is assumed but without checking first so as a result an inconsistent question gives rise to an inconsistent answer.
When AI gets better and stronger AI will tell you this as well 🙂
Or to rephrase: solving a 2d problem in 1d is not correct all the time.
Another rephrase: write a program to calculate square of a natural number? Then input pi, root 2, … complex number, vector, matrix , …
A computer is more-or-less capable of doing all of these things but if working method is limited do not be surprised if program fails or gives dodgy result.
If the above is a bit dodgy well at least it provides a workaround namely: do a consistency test first.
In above consistency twiddles about randomly as syntax in a programing environment, grammar in written and context as in relevant context
@JxH
02/20/2025 - 11:36 PMWe need more Marcus du Sautoy on YouTube.
@mohammadsareh4732
02/20/2025 - 11:36 PMGodel is using man made sentences and try to numerically prove the statement is true, of course he can't. Don't blame mathematics. Science is the struggle of man to know what exists. The only text written to be numerically proven is the Holy Quran by Allah, not by man. Man does not have that ability to write such sentences which can numerically be proven. Even by using the %100 perfect language of Arabic.
گودل از جملات انسان ساخته استفاده می کند و سعی می کند صحت این گفته را به صورت عددی ثابت کند، البته نمی تواند. ریاضیات را سرزنش نکنید علم مبارزه انسان برای شناخت آنچه هست است. تنها متنی که برای اثبات عددی نوشته شده است، قرآن کریم توسط خداوند است، نه توسط انسان. انسان توانایی نوشتن چنین جملاتی را ندارد که به صورت عددی قابل اثبات باشد. حتی با استفاده از 100% زبان کامل عربی
@NomadUrpagi
02/20/2025 - 11:36 PMThis fact makes me a little sad and upset.I felt similarly when discovered that light from some stars will never reach us. Or some lovers will never reunite…
@goedelite
02/20/2025 - 11:36 PMI don't see what is so strange about the existence of statements that are true (or for which no exception can be found) but which cannot be proved within a given system. A situation is comprehensible in which the present system lies within another that is more inclusive and, in fact, includes the unproved statement as provable within it.
@robertferraro236
02/20/2025 - 11:36 PMIt is merely a case of our axioms not being complete. I have a strong feeling that we will discover some of these new axioms in 2025 and that they will lead to proof of Goldbach's Conjecture, Twin Prime Conjecture, and many other deep problems. It is just a hunch.
@Legs-c3o
02/20/2025 - 11:36 PMI’m either way too high or not high enough
@devendrakumarchaturvedi1084
02/20/2025 - 11:36 PM11:00 refering to itself will mean the number is asking to divide me by myself. and if it is asking some other number to ask us that the number is asking be divided by itself.
It is like a prime number having divisors 1 and itself.
Godel statement is the prime number of the number system where trueness of a number is measured by it's divisibility by other numbers.
as we have accepted divisor 1,n dont count as divisors when determining if it prime or not.
so to believe that "this statement true" just because it is saying so will mean he have considered n divides n so it it is not prime.
@KayAmooty433
02/20/2025 - 11:36 PMPart two of silly newbie questions: why does this make me wonder what the link is between Godel's IT and the concept of Superposition?
@KayAmooty433
02/20/2025 - 11:36 PMI'm still struggling, friends. I'm new to this. New to physics and math, so please go easy on me. Here's what I think when I read that statement: "This statement cannot be proved from the axioms that we have for our system of mathematics." A. What statement? Where is the content? B. Don't his findings – when converting this statement devoid of content into Godel Coding – demonstrate an inconsistency between language and math instead of a mathematical incompleteness?
@djayjp
02/20/2025 - 11:36 PMI don't think there's any problem. It's just turning logical, complete, proved statements into nonsensical statements. Much ado about nothing.
@vikineo
02/20/2025 - 11:36 PMIf it can be proved that rh is unprovable within axiomatic system, why would that mean it can’t be false? It could also mean it cannot be true right? Why?
@tapejara1507
02/20/2025 - 11:36 PMHe was just 24 when he cooked this. what a mad lad.
@WarriorOfTheLulz
02/20/2025 - 11:36 PMGödel found a way to represent a formal language in an obscure and convoluted natural language which grammatically shows some correlation with the formal language and integrated an expression into it that violates the rules of the formal language, therefore proving that natural languages are prone to paradoxes.
@zemm9003
02/20/2025 - 11:36 PMProving the Riemann Hypothesis in the way described would actually reveal a fundamental inconsistency in Mathematics because you prove that A is undecidable but then you use that to decide it and thus you prove at the same time that it is not provable and still you find a proof for it.
@skatethe4881
02/20/2025 - 11:36 PM5:08 "… we haven't Gödel of the axioms…"
@RomanBabb
02/20/2025 - 11:36 PMso basically linguistic paradoxes translated into mathenatics and in turn if we cannot prove something weve proven its unprovable and for something like riemanns hypothesis it may be more complicated than a yes or no questions
@dougcane4059
02/20/2025 - 11:36 PMThere are NO 'paradoxes' – there is only a misunderstanding of language.
@scathiebaby
02/20/2025 - 11:36 PMIf the Riemann Hypothesis is unprovable, the unprovabality can't be proven, because proving it would prove the Riemann Hypothesis, making it provable.
@lucelxebinog
02/20/2025 - 11:36 PMI do have a problem with the idea to call something true while not being proven yet. To me this already tells me that there are definition problems at thr start
@OrafuDa
02/20/2025 - 11:36 PM4:12 Actually, what Gödel showed is that for any mathematical system that contains at least predicate logic and the natural numbers, there are sentences that can be expressed within that mathematical system, but which cannot be proved within that system — if the system is consistent, which can also not be proved from within that system itself. — But of course these sentences or some equivalent axiom can then be added to the mathematical system. So, if an equivalent axiom has been added to the mathematical system, then the sentence can also be proved within that system.
So, whenever someone encounters such an unprovable sentence, it can be added to the mathematical system to form a larger mathematical system. And this can be repeated for more unprovable sentences. But no matter how many of these sentences are added as axioms to the mathematical system, Gödel’s incompleteness theorem proved that there will always be more sentences that cannot be proved within the system.
Gödel’s (first) incompleteness proof is not saying that there are statements that will never be proved in any mathematical system. It is saying that for any mathematical system there are always statements that cannot be proved within that specific system … but they may be proved within other systems. So, no consistent mathematical system is complete … but any sentence outside of any system can be proved within other systems that are “better equipped” to prove that sentence.
Side note: I am sidestepping the notion of “truth” here. Of course, people usually want to prove true sentences, and disprove false sentences. I wrote the above paragraphs in such a way that “proving a sentence” can mean “proving a true sentence” (but the truth is not evident from within the system), or “proving that a false sentence is false, ie. disproving the sentence” (but the falsehood of the sentence is not evident from within the system). And one would usually only want to add true sentences to a mathematical system (to maintain its supposed or otherwise proved consistency), which means that the inverse of a false sentence would be added, not the false sentence itself. This side note should be kept in mind while reading the above paragraphs (again). It makes the formulation of the above paragraphs simpler when “truth” is kept out of them, while still holding these ideas in mind.
@matemindak384
02/20/2025 - 11:36 PMI don't understand the issue. Like he said in his example about Riemann Hypothesis, if we know that a theorem is "not proveable" then we know that it must be true (if it was false, then a counterexample must exist –> the theorem isn't unproveable) . That's literally an indirect proof that the said theorem is true, so how can we say that it's unproveable? I'm really confused.
@justsomeone953
02/20/2025 - 11:36 PMI wonder if Gödel thought about "just burn the paper and never publish it" xD
@justsomeone953
02/20/2025 - 11:36 PMHaha
Showing that the Rieman hypothesis does not have a proof proves it is true.
Mind blown
…but how do you find a proof for that? XD
@PsychedelicSilver
02/20/2025 - 11:36 PMGodel did nothing but add the concept of perspective to mathematics. Wrong is wrong and calling it wrong doesn't make it right. Language and math are 2 beautiful ways to describe reality but they translate like English to ancient Greek.
@MrXccv79
02/20/2025 - 11:36 PMit sounds so arbitrary the way giving codes to statements. but I am no mathematician