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Curt Jaimungal (Me): Demystifying Gödel's Theorem - What It Actually Says
May 5, 2025
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The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science, they analyze culture, they analyze finance, economics, business, international affairs across every region.
I'm particularly liking their new insider feature was just launched this month it gives you gives me a front row access to the economist internal editorial debates where senior editors argue through the news with world leaders and policy makers and twice weekly long format shows basically an extremely high quality podcast whether it's scientific innovation or shifting global politics the economist provides comprehensive coverage beyond headlines.
As a TOE listener, you get a special discount. Head over to economist.com slash TOE to subscribe. That's economist.com slash TOE for your discount.
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Many popularizers who use Gödel's incompleteness theorem to make bold claims about fundamental limits of human knowledge have made a category error. Gödel, he concluded that at some point in mathematics, you just have to make something up. Gödel's theorem shows the limitations of almost every theory of reality. My favorite is consciousness. There is a hole at the bottom of math, a hole that means we will never know everything with certainty.
Today I'll cover misuses, why they're misuses, and I'll also talk about what Gödel's theorem actually says because to state it in its catchy glib form misses the necessary rigor required to know its domain of application. The TLDR is that Gödel's incompleteness theorem is about axiomatization, not epistemology. Now there's an asterisk here which I'll get to later. Epistemology is just fancy technical jargon for what and how we can know. So, knowledge. Knowledge.
The gist of girdle is that someone can hand you any concrete, mechanically checkable theory, let's call it F, and girdle's machinery then spits out explicit arithmetic statements, maybe a busy beaver statement, that F, your little machinery here, can neither prove nor disprove. Those sentences set a hard ceiling for that particular proof verifier, even though you yourself can always zoom out, beef up the axioms, and push the line further.
See this excellent exposition here by Scott Aronson on that. As for these other more pop-sci videos, I used to believe these slogans as well. It's difficult not to be seduced by the mysticism around Gödel's theorem that changed for me personally when I was at the University of Toronto.
It was there that I took a course where part of the final exam was to actually prove Gödel's first incompleteness theorem, assuming only a specific model of arithmetic. Now, this was tricky because not only did you have to be acutely clear in each line of reasoning, but you had to properly understand the assumptions. For instance,
What does it mean to be powerful enough to encode arithmetic? What's arithmetic? What's a model? What's an interpretation? What's a universe in model theory? What's the difference between syntax and semantics?
Now, I bring this up not because I'll be assuming you know these, but to emphasize that theorems are all based on terribly tedious assumptions that one can't just gloss over. Gödel was a mathematical genius, one of Einstein's closest friends in his later years. In 1931, Gödel blew apart Hilbert's program in a single stroke. He severed mathematical truth from formal proof. He also fixed exact limits on axiomatization.
He also helped lay the bedrock for computability theory, and he also guaranteed that these undecidable problems will exist forever. And that's all just from one year. Add Gödel's completeness theorem, which yields the compactness principle, also the constructible universe, which proved the consistency of the generalized continuum hypothesis with the axiom of choice.
and then of course his rotating universe solution that he gave to Einstein as a 70th birthday present where there are time travel solutions in general relativity. All of that is fantastic and can't be understated. Now let's get back to that one-liner. Gödel's incompleteness theorem is about axiomatization and not epistemology. Questions in the field of epistemology are questions that deal with the nature and sources and limits of knowledge. The nature of knowledge even includes defining what knowledge is.
This isn't trivial, you can see the Gettier's problem and the recent deep dive with Jennifer Nagel. Link will be in the description. On the other hand, Gödel's incompleteness theorems are regarding what can be proven within a formal system. Now every word here is important. First, notice that there's a plural and that's because there are two incompleteness theorems. The one that most people talk about when they mention his incompleteness theorem is the first one. Actually something you should notice that I keep saying Gödel. That's just my
English pronunciation of his name, you're stuck with it.
Formal has a specific meaning and within also has a specific meaning. The claims of Gödel's theorems are of a different domain than the claims in epistemology. The confusion between this has led to some wild misinterpretations allowing pundits to import metaphysics through a linguistic loophole.
Again, each word here is important. If you have a formal system that's consistent, so it doesn't prove contradictions, and it can't be mechanically checked, so that's the recursive axiomatization, then there will always be true statements capable of being stated within that system, but that can't be proved within the system itself. Neither can you prove its negation. Sounds profound.
However, unless you already think knowledge is a single fixed formal system that we are trapped in Robinson arithmetic, say, then it's difficult to see why this would have any implication on our general limitations or lack thereof of knowledge. But why? Well, for one, we use multiple systems for knowledge generation, not just formal. We also use informal reasoning like intuition. We also use empirical observation.
As an aside here, you may wonder, but Kurt, wait, doesn't Penrose have something to say about this? Doesn't he have an argument in this domain? Yes. And I've spoken to Penrose twice. I'll place links in the description. And at another point on this channel, I'll cover where what I'm about to say in this entire video either agrees with Penrose, diverges from him, or is independent of him, aside over. Now, furthermore,
Unlike what you may infer from Russell's 200 page proof of one plus one equals two, we don't actually use axioms to justify our knowledge. Instead, we use our prior established mathematical knowledge to justify the fittedness of some axiomatic system. So it's directly the opposite. Russell himself even remarked about this. In some sense, using axioms to justify our knowledge would be like building scaffolding to support the ground. Sure, it's impressive.
but entirely unnecessary. Anyhow, even further, we didn't need Gödel to tell us that there exists some facts about math that will forever be beyond our grasp. Why? Because unless you're this metaverse merging Zuckerberg, you have finite working memory, you have finite processing speed, you even have a finite life. So you can't mentally compute some, say, 94,000 digit addition problem, let alone something with Googleplex digits, let alone some exam paper with basic addition problems,
Goethe's theorem shows the limitations of almost every theory of reality. My favorite is consciousness on the monism.
No, not exactly. Gödel shows only that formal, recursively axiomatized systems of arithmetic leave some arithmetic truths unprovable. Nothing in the proof even mentions consciousness, mind-matter dualism, non-dualism, or a purely rational description. As Gregory Chaitin likes to say, truth outruns proof. But that's a fact inside math.
It's not carte blanche for metaphysics. It's important to be clear in one's presentations of mathematical and physics material. Actually, I was scheduled to speak with Deepak in person on this channel, but I told his people respectfully that I would have to question him on some spurious uses of quantum physics if he was to bring that up. And then they canceled the day before the scheduled interview. Note that this wasn't Deepak canceling. It was someone or some people on his team.
Can mathematics be constructed as a completely self-consistent set of rules?
Now this one is an odd one. It sounds like what Neil is talking about is just axioms, which are what you assume to be true. But Gödel didn't show that you have to have axioms. The necessity of axioms is a fundamental prerequisite for the
Kind of formal deductive systems that girdle was analyzing so neil degrasse tyson was accurately describing the nature of axioms in some sense although i would disagree with that we just make anything up but he mistakenly presents this as the core revelation of girdles and completeness theorems also as we mentioned with birch and russell you aren't looking for arbitrary axioms but rather your axioms are justified by their consequences
I've heard Michio Kaku say that Gödel's theorem implies that we'll never know a theory of everything in physics, but this is another misstep. Incompleteness is about proof-theoretic closure, not about differential equations describing nature. Philosopher Graham Appie points out that Gödel places no a priori barrier on how well equations can model empirical data. In other words, Gödel doesn't equal epistemology.
The theorem shows a technical fact about recursively axiomatized formal systems. It's silent on what human beings or physicists can know by other means, such as intuition, experiments, higher order logics, or new axioms, etc. Also, one shouldn't confuse proof with some capital T truth.
Incompleteness is an internal mathematical gap, but it doesn't automatically license metaphysical conclusions about consciousness, God, multiverses, etc. Physics isn't obviously a Gödel-style system. A physical toe may be stochastic or continuous, or even algorithmically infinite. Gödel's diagonal step needs none of these. Physics, after all, is science, not mathematics. We use mathematics because it works, not because we think nature is mathematics.
This in my present deliberations is a correct statement. Physics is model-based and provisional. See my recent video about why the so-called proofs that particles take all possible paths is faulty. There is a hole at the bottom of math, a hole that means we will never know everything with certainty.
No, not exactly Derek. Girdle shows that no single recursively axiomatized system can prove all arithmetic truths. Not that human beings who can move between systems by the way will never know anything for certain. I've heard a popularizer say that Girdle showed that mathematics collapses under its own weight. Math didn't collapse
The girdle's theorem only blocks a complete and consistent single axiom set, not mathematical reasoning as a whole. By the way, a subtle and extremely important point that many seem to miss is that these unprovable truths by girdle, the ones that girdle guarantees via his sentences,
They're not globally true. Now why is that? It's because of Gödel's completeness theorem, which is a different theorem. If a statement is true in all models, then your system necessarily proves it.
and this contradicts its unprovability. Therefore, any undecidable statement is true in some models and false in others. It counts only as true relative to a certain model and it's usually the standard one. So these unprovable truths aren't some stash of universal facts forever beyond reach. They're actually model dependent. Now you may get the impression that, well, these experts, they don't know what they're talking about, but that's not the correct takeaway. These are extremely bright and precise people.
What's funny is that I find most physicists are quite humble and tentative when it comes to ontological or epistemological claims behind closed doors. You can see this on this channel itself, this series of everything podcasts that I have. This is something that I try to elicit because I treat this podcast like office hours as if I'm privately in a professor's office asking questions and I don't care about the audience.
And the professors, they noticed that they just speak to me with the same level of technicality that they would behind closed doors. And the audience is just there like a fly on the wall. It's difficult to do open door physics. But by the way, someone who does do open door physics extremely well is Gabriel Carcassi. But however, Gabriel is rare. And for some reason, when something is framed as this is for a general audience, those popularizers go into this
Orchidaceous Popular Science Communication Mode. They want to convey some voodoo to seduce the audience. Someone said about my previous video about all possible paths. They said like, Kurt, come on. You're fighting a straw man because no physicist actually claims that particles literally explore all possible paths. At Barf Manager, I'm sorry to disappoint you.
And let me use my language carefully. I was going to say it is as if many physicists would say no, it does. So the statement is that the electron explores both routes at the same time at once, let's say on its route from the electron gun through the slits to the screen. Many physicists now would say that that is a correct description of reality.
The political does and this by the way was on a large channel explaining physics to millions of people. I think that's more harmful in the long run since it leads to misconceptions that others have to dispel and then the popularizer of science who came in with those claims feels like they have to defend themselves and then so they do and the audience just sees this conflict and combativeness
It gives credence to those who want to dismiss science or scientists as not knowing what they're talking about claiming hey it's all conjecture etc i think it's more fruitful from the get-go to be honest about what we don't know and where our theories begin and end.
It's more honest at least i think so to spend the extra few minutes to say what you want to say more precisely with the correct caveats now i understand the alternative is to cajole the audience with showy quantum pornography to continue to sell seats or get more views but who cares. Who cares.
If that's at the expense of adding more confusion, or even worse, making the audience feel like they understood something on a deep level when what was presented was superficial with analogies that don't hold up under just two levels of question asking, I would imagine that acknowledging shortcomings or those disastrous openly
Enhances long-term trust by demonstrating a commitment to validity over reputation. Now again, a refrain on this channel is to not wholesale by what anyone is saying, not even me. You have to listen to the arguments and not be swayed by pomp and circumstance. One always has to ask, what is the specific argument? Understand it. When critiquing something, critique that, not the person, not the person's affiliations or lack thereof or the animation or what have you. When critiquing you think, what is the claim?
and the argumentation for that claim or the justification, and that's it.
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Getting back to incompleteness, our inability to grasp most of the theorems of arithmetic or Robinson arithmetic has nothing to do with Gödel and everything to do with our finite cognitive capacities. Even if we somehow overcame these Gödelian limitations, these mundane constraints would still prevent us from knowing many if not most mathematical truths.
Like those high-digit span tests that we talked about earlier. Also, think about this. You know that the Gödel sentence itself is true despite being unprovable within the system. How? Because you can step outside the formal system and use meta-reasoning. You're not that single formal system. This fact alone dismantles the claim that Gödel's theorem imposes fundamental epistemological limits.
You should know though that you have this undecidable truth and you may think okay well why can't i just append it to my axiom set for some given fixed theory that still doesn't erase the ceiling that girdle set it just relocates it because the stronger theory that you've now formed carries its own incompleteness sentence and that just goes on ad infinitum chetan says that rather than girdle giving us limits girdles freed us
Gregory Chaitin, by the way, is the founder of algorithmic information theory, who basically pioneered a whole new field of math when he was just a teenager after being inspired by Gödel. I view it as saying that mathematics is creative and will always be creative. Gödel incomplete that is really good news. It's not bad news. It's not a closed system. It's an open system. Now let's get back to that asterisk that I talked about earlier. It's that every computable proof engine is an axiom system.
So what and what does that mean? Well, Gödel's theorem still marks objective limits on what such an engine can certify. Now the tie to epistemology does come from our tie to relying on these external proof-checkers
for these undecidable theorems. The Paris-Harrington theorem is another example, as it shows there exist statements that are unprovable in piano arithmetic, or what have you, yet are actually proven using other mathematical resources. Therefore, we have knowledge of these truths despite them being unprovable in one formal system. So, as Graham Oppie points out, when theologians compare our knowledge of mathematics to our knowledge of God, claiming both are fundamentally linked by Gurdelian restraints,
They're making an elementary category error. Our mathematical knowledge isn't analogous to our theological knowledge, and Gödel's theorem doesn't support any such analogy.
Likewise, when continental philosophers invoke Gödel to defend notions of radical indeterminacy or undecidability of theories, they're misapplying a precisely defined mathematical result to vaguely articulated philosophical positions. Now someone asked me, but Kurt, doesn't Gödel imply that our knowledge is partial? And I'll just summarize this conversation. I'll put a link on screen. I asked in response,
What do you mean by partial? Do you mean finite, like some decidable theory? And the conversation partner said, yes. And I then went on to explain that Gödel relies on an infinite arithmetic. Otherwise, every theorem could be listed in full because the set would be finite and membership then could be mechanically decided by straightforward brute force inspection. This person was then saying, okay, so then should we not use Gödel's theorem as a simile for knowledge?
And then my claim is that knowledge is quite slippery here. The point is that the girdle sentence is constructed using the assumption that the underlying formalism contains the whole of elementary arithmetic, meaning you can always form numerals of unbounded size and prove addition and multiplication facts for each of them, etc. And that unboundedness is the infinity that's baked into the proof. So if you impose a hard bound on numeral size, then the whole incompleteness argument never even starts.
Even if our minds were formal systems, no one is consistent in their beliefs. Thus, Gödel's theorem wouldn't apply to your knowledge generation anyhow if you're basing your generation on some formal analogy with belief as axioms since, recall, one of the assumptions of Gödel's theorem was consistency to begin with.
Now, it turns out, axiomatization does not equal knowledge. To me, that's a huge takeaway from Gödel, although you arguably could have concluded that from the Lohenheim-Skollam theorems a few years before. I wrote a substack post about this and I'll place a link on screen and in the description.
Note that I worked on this video on and off for a few weeks writing the script and I tried to simplify it, but I decided it was better to be honest with you about the complexity and tell you the details along with signposts and references for you to learn more than to try to wow you with some cursory response that may leave you saying, whoa, my mind is blown. But then a day later, you're left scratching your head like
wait what did i actually learn the overall lesson is that girdles and completeness theorem is an extremely specific theorem with extremely specific assumptions that are
Quite thorny to untangle, Gödel sharpened vagueness by pinpointing concrete arithmetic statements that no fixed, consistent, computable axiom set can nail down. The relationship between formal systems and human cognition isn't straightforward, and while we aren't identical to current formal systems, the extent to which human reasoning can be modeled computationally is nowhere near settled. See the church-turing thesis applied to mind, for instance.
To draw epistemological consequences from Gödel requires teasing out each of these assumptions and much of the time, such as the quotations that I've cited before, when people say that Gödel showed that there's some human limitation, you have to pause and decode because the decoded version is most often a resounding no, not exactly.
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"text": " No, not exactly. Gödel shows only that formal, recursively axiomatized systems of arithmetic leave some arithmetic truths unprovable. Nothing in the proof even mentions consciousness, mind-matter dualism, non-dualism, or a purely rational description. As Gregory Chaitin likes to say, truth outruns proof. But that's a fact inside math."
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"text": " It's not carte blanche for metaphysics. It's important to be clear in one's presentations of mathematical and physics material. Actually, I was scheduled to speak with Deepak in person on this channel, but I told his people respectfully that I would have to question him on some spurious uses of quantum physics if he was to bring that up. And then they canceled the day before the scheduled interview. Note that this wasn't Deepak canceling. It was someone or some people on his team."
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{
"end_time": 600.333,
"index": 24,
"start_time": 581.886,
"text": " Can mathematics be constructed as a completely self-consistent set of rules?"
},
{
"end_time": 615.486,
"index": 25,
"start_time": 600.333,
"text": " Now this one is an odd one. It sounds like what Neil is talking about is just axioms, which are what you assume to be true. But Gödel didn't show that you have to have axioms. The necessity of axioms is a fundamental prerequisite for the"
},
{
"end_time": 637.381,
"index": 26,
"start_time": 615.486,
"text": " Kind of formal deductive systems that girdle was analyzing so neil degrasse tyson was accurately describing the nature of axioms in some sense although i would disagree with that we just make anything up but he mistakenly presents this as the core revelation of girdles and completeness theorems also as we mentioned with birch and russell you aren't looking for arbitrary axioms but rather your axioms are justified by their consequences"
},
{
"end_time": 661.886,
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"start_time": 637.381,
"text": " I've heard Michio Kaku say that Gödel's theorem implies that we'll never know a theory of everything in physics, but this is another misstep. Incompleteness is about proof-theoretic closure, not about differential equations describing nature. Philosopher Graham Appie points out that Gödel places no a priori barrier on how well equations can model empirical data. In other words, Gödel doesn't equal epistemology."
},
{
"end_time": 681.493,
"index": 28,
"start_time": 662.244,
"text": " The theorem shows a technical fact about recursively axiomatized formal systems. It's silent on what human beings or physicists can know by other means, such as intuition, experiments, higher order logics, or new axioms, etc. Also, one shouldn't confuse proof with some capital T truth."
},
{
"end_time": 711.544,
"index": 29,
"start_time": 681.903,
"text": " Incompleteness is an internal mathematical gap, but it doesn't automatically license metaphysical conclusions about consciousness, God, multiverses, etc. Physics isn't obviously a Gödel-style system. A physical toe may be stochastic or continuous, or even algorithmically infinite. Gödel's diagonal step needs none of these. Physics, after all, is science, not mathematics. We use mathematics because it works, not because we think nature is mathematics."
},
{
"end_time": 730.162,
"index": 30,
"start_time": 711.613,
"text": " This in my present deliberations is a correct statement. Physics is model-based and provisional. See my recent video about why the so-called proofs that particles take all possible paths is faulty. There is a hole at the bottom of math, a hole that means we will never know everything with certainty."
},
{
"end_time": 752.449,
"index": 31,
"start_time": 730.811,
"text": " No, not exactly Derek. Girdle shows that no single recursively axiomatized system can prove all arithmetic truths. Not that human beings who can move between systems by the way will never know anything for certain. I've heard a popularizer say that Girdle showed that mathematics collapses under its own weight. Math didn't collapse"
},
{
"end_time": 770.435,
"index": 32,
"start_time": 752.449,
"text": " The girdle's theorem only blocks a complete and consistent single axiom set, not mathematical reasoning as a whole. By the way, a subtle and extremely important point that many seem to miss is that these unprovable truths by girdle, the ones that girdle guarantees via his sentences,"
},
{
"end_time": 783.012,
"index": 33,
"start_time": 770.435,
"text": " They're not globally true. Now why is that? It's because of Gödel's completeness theorem, which is a different theorem. If a statement is true in all models, then your system necessarily proves it."
},
{
"end_time": 811.971,
"index": 34,
"start_time": 783.387,
"text": " and this contradicts its unprovability. Therefore, any undecidable statement is true in some models and false in others. It counts only as true relative to a certain model and it's usually the standard one. So these unprovable truths aren't some stash of universal facts forever beyond reach. They're actually model dependent. Now you may get the impression that, well, these experts, they don't know what they're talking about, but that's not the correct takeaway. These are extremely bright and precise people."
},
{
"end_time": 835.299,
"index": 35,
"start_time": 812.312,
"text": " What's funny is that I find most physicists are quite humble and tentative when it comes to ontological or epistemological claims behind closed doors. You can see this on this channel itself, this series of everything podcasts that I have. This is something that I try to elicit because I treat this podcast like office hours as if I'm privately in a professor's office asking questions and I don't care about the audience."
},
{
"end_time": 862.108,
"index": 36,
"start_time": 835.64,
"text": " And the professors, they noticed that they just speak to me with the same level of technicality that they would behind closed doors. And the audience is just there like a fly on the wall. It's difficult to do open door physics. But by the way, someone who does do open door physics extremely well is Gabriel Carcassi. But however, Gabriel is rare. And for some reason, when something is framed as this is for a general audience, those popularizers go into this"
},
{
"end_time": 883.78,
"index": 37,
"start_time": 862.398,
"text": " Orchidaceous Popular Science Communication Mode. They want to convey some voodoo to seduce the audience. Someone said about my previous video about all possible paths. They said like, Kurt, come on. You're fighting a straw man because no physicist actually claims that particles literally explore all possible paths. At Barf Manager, I'm sorry to disappoint you."
},
{
"end_time": 909.241,
"index": 38,
"start_time": 884.224,
"text": " And let me use my language carefully. I was going to say it is as if many physicists would say no, it does. So the statement is that the electron explores both routes at the same time at once, let's say on its route from the electron gun through the slits to the screen. Many physicists now would say that that is a correct description of reality."
},
{
"end_time": 930.333,
"index": 39,
"start_time": 909.531,
"text": " The political does and this by the way was on a large channel explaining physics to millions of people. I think that's more harmful in the long run since it leads to misconceptions that others have to dispel and then the popularizer of science who came in with those claims feels like they have to defend themselves and then so they do and the audience just sees this conflict and combativeness"
},
{
"end_time": 945.674,
"index": 40,
"start_time": 930.333,
"text": " It gives credence to those who want to dismiss science or scientists as not knowing what they're talking about claiming hey it's all conjecture etc i think it's more fruitful from the get-go to be honest about what we don't know and where our theories begin and end."
},
{
"end_time": 966.749,
"index": 41,
"start_time": 945.674,
"text": " It's more honest at least i think so to spend the extra few minutes to say what you want to say more precisely with the correct caveats now i understand the alternative is to cajole the audience with showy quantum pornography to continue to sell seats or get more views but who cares. Who cares."
},
{
"end_time": 986.101,
"index": 42,
"start_time": 967.039,
"text": " If that's at the expense of adding more confusion, or even worse, making the audience feel like they understood something on a deep level when what was presented was superficial with analogies that don't hold up under just two levels of question asking, I would imagine that acknowledging shortcomings or those disastrous openly"
},
{
"end_time": 1016.101,
"index": 43,
"start_time": 986.374,
"text": " Enhances long-term trust by demonstrating a commitment to validity over reputation. Now again, a refrain on this channel is to not wholesale by what anyone is saying, not even me. You have to listen to the arguments and not be swayed by pomp and circumstance. One always has to ask, what is the specific argument? Understand it. When critiquing something, critique that, not the person, not the person's affiliations or lack thereof or the animation or what have you. When critiquing you think, what is the claim?"
},
{
"end_time": 1020.418,
"index": 44,
"start_time": 1016.493,
"text": " and the argumentation for that claim or the justification, and that's it."
},
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"text": " This episode is brought to you by State Farm. Listening to this podcast? Smart move. Being financially savvy? Smart move. Another smart move? Having State Farm help you create a competitive price when you choose to bundle home and auto. Bundling. Just another way to save with a personal price plan. Like a good neighbor, State Farm is there. Prices are based on rating plans that vary by state. Coverage options are selected by the customer. Availability, amount of discounts and savings, and eligibility vary by state."
},
{
"end_time": 1081.852,
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"text": " Close your eyes, exhale, feel your body relax, and let go of whatever you're carrying today. Well, I'm letting go of the worry that I wouldn't get my new contacts in time for this class. I got them delivered free from 1-800-CONTACTS. Oh my gosh, they're so fast. And breathe. Oh, sorry. I almost couldn't breathe when I saw the discount they gave me on my first order. Oh, sorry. Namaste. Visit 1-800-CONTACTS.COM today to save on your first order."
},
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"text": " Getting back to incompleteness, our inability to grasp most of the theorems of arithmetic or Robinson arithmetic has nothing to do with Gödel and everything to do with our finite cognitive capacities. Even if we somehow overcame these Gödelian limitations, these mundane constraints would still prevent us from knowing many if not most mathematical truths."
},
{
"end_time": 1129.974,
"index": 48,
"start_time": 1104.94,
"text": " Like those high-digit span tests that we talked about earlier. Also, think about this. You know that the Gödel sentence itself is true despite being unprovable within the system. How? Because you can step outside the formal system and use meta-reasoning. You're not that single formal system. This fact alone dismantles the claim that Gödel's theorem imposes fundamental epistemological limits."
},
{
"end_time": 1155.179,
"index": 49,
"start_time": 1129.974,
"text": " You should know though that you have this undecidable truth and you may think okay well why can't i just append it to my axiom set for some given fixed theory that still doesn't erase the ceiling that girdle set it just relocates it because the stronger theory that you've now formed carries its own incompleteness sentence and that just goes on ad infinitum chetan says that rather than girdle giving us limits girdles freed us"
},
{
"end_time": 1183.677,
"index": 50,
"start_time": 1155.538,
"text": " Gregory Chaitin, by the way, is the founder of algorithmic information theory, who basically pioneered a whole new field of math when he was just a teenager after being inspired by Gödel. I view it as saying that mathematics is creative and will always be creative. Gödel incomplete that is really good news. It's not bad news. It's not a closed system. It's an open system. Now let's get back to that asterisk that I talked about earlier. It's that every computable proof engine is an axiom system."
},
{
"end_time": 1200.094,
"index": 51,
"start_time": 1183.882,
"text": " So what and what does that mean? Well, Gödel's theorem still marks objective limits on what such an engine can certify. Now the tie to epistemology does come from our tie to relying on these external proof-checkers"
},
{
"end_time": 1230.367,
"index": 52,
"start_time": 1200.452,
"text": " for these undecidable theorems. The Paris-Harrington theorem is another example, as it shows there exist statements that are unprovable in piano arithmetic, or what have you, yet are actually proven using other mathematical resources. Therefore, we have knowledge of these truths despite them being unprovable in one formal system. So, as Graham Oppie points out, when theologians compare our knowledge of mathematics to our knowledge of God, claiming both are fundamentally linked by Gurdelian restraints,"
},
{
"end_time": 1240.998,
"index": 53,
"start_time": 1230.367,
"text": " They're making an elementary category error. Our mathematical knowledge isn't analogous to our theological knowledge, and Gödel's theorem doesn't support any such analogy."
},
{
"end_time": 1266.152,
"index": 54,
"start_time": 1241.288,
"text": " Likewise, when continental philosophers invoke Gödel to defend notions of radical indeterminacy or undecidability of theories, they're misapplying a precisely defined mathematical result to vaguely articulated philosophical positions. Now someone asked me, but Kurt, doesn't Gödel imply that our knowledge is partial? And I'll just summarize this conversation. I'll put a link on screen. I asked in response,"
},
{
"end_time": 1295.879,
"index": 55,
"start_time": 1266.408,
"text": " What do you mean by partial? Do you mean finite, like some decidable theory? And the conversation partner said, yes. And I then went on to explain that Gödel relies on an infinite arithmetic. Otherwise, every theorem could be listed in full because the set would be finite and membership then could be mechanically decided by straightforward brute force inspection. This person was then saying, okay, so then should we not use Gödel's theorem as a simile for knowledge?"
},
{
"end_time": 1323.08,
"index": 56,
"start_time": 1296.288,
"text": " And then my claim is that knowledge is quite slippery here. The point is that the girdle sentence is constructed using the assumption that the underlying formalism contains the whole of elementary arithmetic, meaning you can always form numerals of unbounded size and prove addition and multiplication facts for each of them, etc. And that unboundedness is the infinity that's baked into the proof. So if you impose a hard bound on numeral size, then the whole incompleteness argument never even starts."
},
{
"end_time": 1343.865,
"index": 57,
"start_time": 1324.991,
"text": " Even if our minds were formal systems, no one is consistent in their beliefs. Thus, Gödel's theorem wouldn't apply to your knowledge generation anyhow if you're basing your generation on some formal analogy with belief as axioms since, recall, one of the assumptions of Gödel's theorem was consistency to begin with."
},
{
"end_time": 1360.145,
"index": 58,
"start_time": 1343.865,
"text": " Now, it turns out, axiomatization does not equal knowledge. To me, that's a huge takeaway from Gödel, although you arguably could have concluded that from the Lohenheim-Skollam theorems a few years before. I wrote a substack post about this and I'll place a link on screen and in the description."
},
{
"end_time": 1384.394,
"index": 59,
"start_time": 1360.145,
"text": " Note that I worked on this video on and off for a few weeks writing the script and I tried to simplify it, but I decided it was better to be honest with you about the complexity and tell you the details along with signposts and references for you to learn more than to try to wow you with some cursory response that may leave you saying, whoa, my mind is blown. But then a day later, you're left scratching your head like"
},
{
"end_time": 1394.377,
"index": 60,
"start_time": 1384.667,
"text": " wait what did i actually learn the overall lesson is that girdles and completeness theorem is an extremely specific theorem with extremely specific assumptions that are"
},
{
"end_time": 1422.073,
"index": 61,
"start_time": 1394.599,
"text": " Quite thorny to untangle, Gödel sharpened vagueness by pinpointing concrete arithmetic statements that no fixed, consistent, computable axiom set can nail down. The relationship between formal systems and human cognition isn't straightforward, and while we aren't identical to current formal systems, the extent to which human reasoning can be modeled computationally is nowhere near settled. See the church-turing thesis applied to mind, for instance."
},
{
"end_time": 1444.514,
"index": 62,
"start_time": 1422.381,
"text": " To draw epistemological consequences from Gödel requires teasing out each of these assumptions and much of the time, such as the quotations that I've cited before, when people say that Gödel showed that there's some human limitation, you have to pause and decode because the decoded version is most often a resounding no, not exactly."
},
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"text": " Think Verizon, the best 5G network is expensive? Think again. Bring in your AT&T or T-Mobile bill to a Verizon store today and we'll give you a better deal. Now what's it do with your unwanted bills? Ever seen an origami version of the Miami Bull? Jokes aside, Verizon has the most ways to save on phones and plans where you can get a single line with everything you need."
},
{
"end_time": 1475.111,
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"start_time": 1464.053,
"text": " Local Miami Verizon store today, and we'll give you a better deal. Rankings based on root metrics, root score report dated 1H2025. Your results may vary. Must provide a post-patients or mobile bill dated within the past 45 days. Bill must be in the same name as the person reviewing the deal. Additional terms apply."
}
]
}No transcript available.