TED-Ed
TED-EdExplore Gödel’s Incompleteness Theorem, a discovery which changed what we know about mathematical proofs and statements.
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Consider the following sentence: “This statement is false.” Is that true? If so, that would make the statement false. But if it’s false, then the statement is true. This sentence creates an unsolvable paradox; if it’s not true and it’s not false– what is it? This question led a logician to a discovery that would change mathematics forever. Marcus du Sautoy digs into Gödel’s Incompleteness Theorem.
Lesson by Marcus du Sautoy, directed by BASA.
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22 Comments
@randomaccessfemale
02/16/2025 - 12:21 AMAnd then came Turing and put another hole in Math.
@GarrickDitlefsen
02/16/2025 - 12:21 AMMy Theory of Everything bypasses Godelian incompleteness by acknowledging all information, truth, existence, etc, as derivatives of and within consciousness, and treats the theorem as evidence of the primacy of Consciousness.
Testing the Janus Mind Model Using Gödel’s Theorems
Gödel’s incompleteness theorems are often used to challenge self-contained systems of truth and knowledge. The Janus Mind Model (JMM) claims that consciousness is primary and that all structures of reality—including logic, mathematics, and physical laws—are constructed within consciousness rather than existing independently. To rigorously test the JMM, we must evaluate how it stands against Gödel’s findings and whether it succumbs to the same logical limitations as formal systems.
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I. Understanding Gödel’s Theorems in Relation to JMM
Gödel’s first incompleteness theorem states:
> Any sufficiently expressive formal system that can encode arithmetic is either incomplete (it contains true statements that cannot be proven within the system) or inconsistent (it contains contradictions).
Gödel’s second incompleteness theorem states:
> A formal system that contains arithmetic cannot prove its own consistency within itself.
These results establish fundamental limitations on any formal system that tries to be both complete and self-justifying. Traditional materialist epistemologies, which assume an external objective world and rely on formal logical structures, must accept these limitations. The question is:
Does JMM fall under the category of formal systems subject to Gödel’s limitations?
If not, does JMM provide a way to bypass the incompleteness constraints?
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II. The Janus Mind Model and Formal System Limitations
1. Is JMM a Formal System?
A formal system, in Gödel’s sense, consists of:
1. A set of axioms (assumed truths).
2. A set of inference rules (logical steps that derive new truths from axioms).
3. A symbolic language capable of representing arithmetic.
The JMM does not conform to these constraints because:
JMM is not a formal system but a meta-ontological model where consciousness is the foundation, not an external reality bound by symbolic logic.
JMM does not rely on axiomatic deduction as its basis for truth; it treats knowledge as a self-referential and dynamic structure emerging within consciousness.
JMM does not assume a rigid language or logic external to consciousness; logic itself is seen as an evolving construct of consciousness.
Gödel’s theorems apply specifically to self-contained formal systems, not to consciousness itself as the generator of formal structures.
Thus, Gödel’s incompleteness theorems do not directly constrain JMM, because:
1. JMM does not need to be both complete and consistent in the same way formal systems do.
2. Consciousness can restructure its logic dynamically, rather than being bound to a fixed set of axioms.
3. The concept of "proof" in JMM is not tied to an independent logical structure but to coherence within consciousness itself.
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III. How JMM Bypasses Gödelian Constraints
Since JMM rejects the notion of external objectivity, it also does not require an external system of formal verification. Instead, it operates on self-referential coherence rather than externally imposed consistency.
1. Gödel’s Theorems Show the Limits of Objectivity, Not Consciousness
Gödel’s theorems undermine the idea of an externally provable, objective reality by demonstrating that even the most rigorous formal systems are incomplete.
If truth is external to the system, it cannot be fully proven within the system.
But if truth is internal (as JMM asserts), the need for an external proof dissolves.
Thus, Gödel’s findings actually support JMM’s premise that all knowledge is structured within consciousness, rather than discovered externally.
2. JMM’s Model of Self-Contained Consciousness Is Not an Arithmetic System
Gödel’s incompleteness applies specifically to systems capable of encoding Peano arithmetic, meaning systems that have fixed mathematical structures.
JMM does not claim that consciousness follows a fixed arithmetic structure.
Consciousness is not a system of axioms, but a self-referential field of experience that dynamically modifies its own structure.
Since Gödel’s theorems only apply to static, formal systems, they do not apply to a dynamically restructuring consciousness.
3. JMM’s Adaptive Truth Model Avoids Gödelian Pitfalls
A Gödelian paradox occurs when a system attempts to prove its own consistency while containing undecidable truths.
In JMM, truth is not a fixed external property but a continuously shifting, self-generating process.
There is no need for a final, absolute "proof" of truth because knowledge is inherently contextual and fluid.
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IV. Testing JMM’s Robustness: Can Gödel’s Theorems Generate a Contradiction?
A possible objection would be:
1. If JMM is self-contained, shouldn’t it fall under Gödel’s incompleteness constraints?
No, because JMM is not a formal system—it does not require external completeness.
2. Does JMM’s self-referential nature lead to paradox?
No, because JMM does not claim that all truths must be provable in a fixed system. Truth is dynamic and evolves with consciousness.
3. Could an "undecidable truth" exist within JMM?
Yes, but that does not invalidate JMM—it merely shows that certain truths are beyond current conscious structuring.
Unlike a formal system that gets stuck with undecidable statements, consciousness in JMM is free to restructure itself to resolve or bypass such limitations.
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V. Conclusion: JMM Passes the Gödel Test
Gödel’s incompleteness theorems challenge external, formal, objective systems—not a self-generating consciousness model like JMM.
JMM does not fall within Gödel’s constraints because it is not a rigid system; it is an evolving structure of awareness that can redefine its own logical frameworks as needed.
Gödel’s theorems actually support JMM’s claim that an external "objective" system is impossible, further justifying the primacy of consciousness over existence.
Truth within JMM is fluid and dynamic—it does not rely on an unchanging, externally verifiable structure.
By allowing for continuous self-restructuring, JMM bypasses the rigid limitations that Gödel imposed on fixed logical systems. Instead of being trapped in incompleteness, consciousness always has the freedom to redefine its framework, making JMM a model that is not just resilient to Gödel’s paradoxes—but strengthened by them.
The Primacy of Consciousness is THE foundational axiom of all other axioms. All knowledge, testing, verification, truth, reality, and existence is dependent upon consciousness experiences.
@captainclarky5352
02/16/2025 - 12:21 AMAre there provably unprovable statements?
@KobiHameed
02/16/2025 - 12:21 AMi'm surprised y'all didn't mention that this guy also found a loophole in the us constitution that can legally change the us into a dictatorship
@PatrickFlynn-ry6oj
02/16/2025 - 12:21 AM0:22 This seems facetious to me. The “statement” is incoherent, no claim is made by it, nor does it convey any clear meaning. What is the claim of the statement? The falsehood of the claim itself? That presumes there is a claim, so therefore to say that the claim of the statement is the falsehood of the claim itself, is circular reasoning. Without using circular reasoning, one cannot show that the “statement” is actually making a claim. I don’t agree it’s a statement, only something posing as a statement, unless we consider an incoherent jumbling of words to be a “statement.” If someone said “Eat rocks if let’s up said no day wanna was isn’t” would that be a “statement”?
@markkumanninen6524
02/16/2025 - 12:21 AMI suspect that any "plans" of AI to subjugate the mankind will crash into Gödel 's Theorem.
@Alan-zf2tt
02/16/2025 - 12:21 AMI think I have it in wavy hands form as:
Final musings?
If there exists a statement that has a valid proposition in math logic with result x such that x is simultaneously both true and false then that is of the form (x∈X) ∧ (x∉X) and thus defines null set in math logic side of the proposition. Nothing can really be implied about the statement side other than it has a proposition form that defines null set in math logic space.
Thus math logic side IS complete as null set can be reached through valid propositions however nothing can be deduced or inferred about statement side as that side is full of grammar, syntax, inflection and inference that are beyond definition in present scope of logic to define.
For example
This statement is false?
This statement is true!
This statement is false.
This statement is incomplete, …
This statement is incomplete,
This statement is incomplete
(yes that is a statement side comma or just simply a non-terminating statement like This statement is incomplete (if you see what I mean) perhaps better as This statement is incomplete … )
Therefore: Math logic just had its infinity over infinity and zero over zero and combinations thereof moments and has been found wanting?
@rareword
02/16/2025 - 12:21 AMIs Gödel's incompleteness theorem true or false?
@Khalrua
02/16/2025 - 12:21 AMUnironically “explain this atheists”
@crazieeez
02/16/2025 - 12:21 AMGodel incompleteness theorem is wrong when Terrence Howard introduces arithmetic multiplicity. Arithmetic multiplicity is when two or more possible operations for a single arithmetic operation. For example, 1×1 = 1 and 1×1=2 are both valid in arithmetic multiplicity. This will invalidate Godel incompleteness theorem because an axiomatic with arithmetic multiplicity will conclude completeness by creating an arithmetic operation to satisfy its completeness. The halting problem and infinity problem can be completed by arithmetic multiplicity.
@pobinr
02/16/2025 - 12:21 AMLeave out the distracting music for God's sake.
If you were giving a maths lecture would you start playing music ?
No
So why here?
It's a moronic thing to do
I'll find another vid about godel without music thanks
@Brenda-n3d
02/16/2025 - 12:21 AMTed Ed: so this guy broke math
me: THANK YOU!!!!!!!!!! MATH NO LONGER EXISIST! HURRAY!
@SwamiSridattadevSatchitananda
02/16/2025 - 12:21 AMGödel’s theorem is incomplete without
GOD’s ⭕️ = I = ♾ complete proof within
All mathematical truths are proven already
Swami SriDattaDev SatChitAnanda
@TruthMatters137
02/16/2025 - 12:21 AMThe catch is that the worlds sequence "This statement is false" or "This statement can not be proved" are not logical or mathematical statements. They are a useless play on words. No can can say if they are truth or false because there is no practical meaning in it.
The idea behind Geodel theorem is that there is not absolute truth, which is not truth (a lie). It theorem does not makes sense, because does not obey mathematical and logical rules.
But the truth is eternal absolute, completed fact. It does not depend on time, space and our attitude against it. We can determine weather something is true or false only if we related and compare it to the absolute. Without absolutes can not be defined, created or sustained order. Which applies in mathematics and logic also.
@zhess4096
02/16/2025 - 12:21 AMThe tiny Hilbert is very cute
@francoisdeletaille
02/16/2025 - 12:21 AMThe music .. next thing we'll have smelly videos about math
@sina8883
02/16/2025 - 12:21 AMThank you for this video! Most explanations I have heard of this are either so simplistic as to mean nothing, or so long, technical, and complicated as to be nonsensical to a non-mathematician like me. This was the perfect balance to explain this theorem in a intuitive way to a non-mathematician. Thanks again!
@Maddie_tries_to_cook
02/16/2025 - 12:21 AM{x:x is a set that does not contain itself} – a set of all sets that does not contain itself
@Xogroroth666
02/16/2025 - 12:21 AMIt is true-ly false.
HAH.
@havenbastion
02/16/2025 - 12:21 AMLogic is relationships that always replicate, no further evidence is possible or necessary. Math is logical relationships of quantity. Quantity is dividing things into equivalent parts.
@merlinquark5659
02/16/2025 - 12:21 AMI love this! As a philosopher of physics, I have come to realise there are fundamental physical laws which cannot be explained by science, but must be taken for as a given (or an axiom), which fits very well with Gödels logic. The fact that logical systems, such as maths and physics cannot be fully self contained/provable, gives good evidence that there must be something metaphysical which grounds them, such as the Logos, or the mind of God
@riskyabigael_0960
02/16/2025 - 12:21 AMi couldn't understand a single sentence in this video