It doesn't detract from the video, but for the pedants, I'll point out that there are various points in the video where the logic depends on the hidden assumption that system of axions is consistent. In particular, the claim that a statement implying a contradiction means that the statement is unprovable depends on the system being consistent.
Oh this makes sense. What you're saying is that it is similar to the parallel postulate or ZFC? Where something is unprovable and not unprovable from a given set of axioms because it is possible to construct a system where it can work and where it doesn't from said axioms?
I think the case of the parallel postulate is more interesting because it's more similar to actual ZFC modelling in a way since you can, for example, directly construct spherical geometry FROM the geometric axioms, which doesn't satisfy the parallel postulate
This made things click in my brain! Multiple examples/realizations of a definition with different properties are extremely common in mathematics. There being different models of ZFC is only slightly weirder than that there are several different vector spaces. Gödel simply proved can NEVER pin things down to a single "true" model...
Gödel proved that there are unprovable true statements only for sufficiently powerful mathematical systems. He used Peano Arithmetic for his proof, and his proof applies to any system that is ω-consistent. Later extensions proved that it is true for any consistent system (which doesn't have much effect because almost all material systems that are consistent are also ω-consistent). It is possible for simpler systems (like those that lack the concept of a "next number", or that have no reliable way to go from x to x+1) to be complete and consistent. Of course, such systems have extremely limited applicability.
this channel is gold, love the digestable videos, especially for someone like me with only an applied mathematics background (i barely know any pure math)
excellent and very accessible explanation of these concepts! as a bonus it also made it a lot clearer for me what the difference is between the single and double turnstiles, despite you not explicitly explaining them. thanks!
I just proved for odd q that (2^a(q-1/2) -1)/q that it is always an integer when a(x) is the sequence A002326 where a(0) is 1. This can be used to easily check Mersenne primes by setting it equal to 1. Sometimes it breaks like for M11 because multiple values of q work. To fix this, use the smallest value of 2^a(q-1/2) -1 for a chosen q. If it is 1, it is prime. If it isn’t, it is not prime.
I wanna thank you for popping up at the right time. As a physics student who's just completed my sophomore year, and started getting into math, this is both just advanced and simple enough for me to actually be able to learn from. Keep up the good work.
If i summarize correctly, you make two models which are "subset" (?) of the model you want to use, then make them each prove a statement and its negation. This makes the statement unprovable and undisprovable. If so, good job for the video!
Close, but these models aren't subsets of each other. You have a theory, and a statement you're interested in. To prove the statement is independent of the theory, you need two models of the theory (meaning all axioms of the theory hold in the model), where the statement is true in one model, and false in the other model.
@@SheafificationOfG "model of the theory" means the model is subset of the theory right? Or whatever the definition is where the model is the original theory + some additional axioms.
@@HazhMcMoor A theory is a set of rules, whereas a model of a theory is an actual thing that satisfies those rules. The integers do not form a "subset" of the axioms of an integral domain.
This has been hinted at in another comment (I think?) but your demonstration at 8:56 only illustrates that our ring must have characteristic 2, not the much stronger claim that it's Z/2Z
Right, if you just take the axioms of an integral domain and tack on "2 =0" you get integral domains of characteristic 2. I didn't stress the point, but I mentiom prior to the construction that the idea behind "forcing" starts with picking an existing model. So, in my case, I was starting with the model Z (usual integers) and then asserting 2 = 0. I was trying to mirror Cohen's approach to demonstrating the unprovability of CH: he started with a model where CH was true, and then used forcing to extend it to a model where CH failed.
One thing I’m still a bit confused on is how this all plays with the law of excluded middle. If we take it as axiom, then these statements have truth value. Should be think of them being simultaneously true and false (until we create a model where we have chosen a specific value)?
The reasoning at 8:16 feels wrong - To me this sounds like “This statement is disprovable if we add this one rule, so if we remove this rule, we are no longer able to verify it”, like, if I want to prove if Goldbach Conjecture is unprovable, then I add a rule say that “Goldbach is false”, which it obviously is in that ruleset, and then I remove that rule, thus complete the proof that Goldbach is unprovable? Thank you.
See pinned comment. You can add a rule "Goldbach is false", but then you also need your new set of rules to be consistent. And proving it is a hard task. With ring of modulo 2 it's easy - there are only 2 elements. You can check that this new rule never contradicts any previous ones, but I fear proving that your new rulesets with "Goldbach is true"/"Goldbach is false" rules are consistent would be as hard as actually trying to prove/disprove it.
Why did you use the one thinker statue that got blown up? Genuine question. Also great vid, love your content from the few minutes I have seen of it :)
@@SheafificationOfG no you didn't at all. Oh I'm sorry, it is sometimes hard for me to recognize the difference between Canadian accents and US accents. I guess people are right in saying assumptions can make a fool out of you.
Interesting! Are there any other commonly used axiomatic systems (other than ZFC, I mean) of set theory where the continuum hypothesis *can* be proved (or disproved)? And what do we get, if we just add the trueness or falseness of the continuum hypothesis as yet another axiom to the existing ZFC axioms? We should get two different mathematical universes, right? Then maybe we could just explore both of them and use the one that we like better :-D ...I'm pretty sure, this must have been done already?
There are some interesting consequences of CH or its negation (for example: if the continuum hypothesis is true, then there exists a function f(x, y) where the integrals f(x,y)dxdy and f(x,y)dydx over the unit square give different values; in general, the existence of such a function is independent of ZFC, but I'm not sure if it's equivalent to CH). However, there hasn't been anything so earth-shattering that makes the mathematics community at large decide if CH is true or otherwise.
@@SheafificationOfG Wow - that's interesting! Can you point me to any resources where I can learn more about this fact? In that case, I think, I'd opt for the math universe where the CH is false. ...I kinda *want* these two integrals to give the same value - that somehow seems to be "intuitively" true to me. I can "see" it - we are computing the volume between the graph of f(x,y) and the xy-plane in both cases - that should not depend on how we compute it ....at least for continuous functions. For more erratic functions...like - say - some sort of 2D version of the Dirichlet function - hmm... I don't know if the integrals *should* still be equal...maybe not...dunno
@@MusicEngineeer Sorry, I lied to you: the existence of a function where the two iterated integrals over the unit square differ is not *equivalent* to CH, but it is a consequence of CH; I've edited the original comment. You can get more information starting from en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC#Measure_theory and following references, I imagine. (The whole article has a bunch of independence results, some of which are tied to CH, so it's also an interesting read).
I've usually taken unproveable and undisprovable to mean that we can't determine whether something is true, but is decidedly either trrue or false - we just can't prove that - but this video seems to suggest instead that the problem simply has no inherent truth or falsity at all. Am I understanding correctly?
In mathematics, “true” and “false” are relative to the axioms we work with. The continuum hypothesis is neither true nor false from ZFC, because its axioms simply do not force an answer. This means we can extend ZFC to make it either true or false. Basically, answering the question requires a new axiom, and valid axioms exist to give either answer. In science, we consider something “true” if it correctly predicts the behaviour of the universe. This is what most people mean when they say “X is true”. Technically, these two definitions of “truth” are _entirely independent!_ Mathematics is not formally obliged to match reality. However, the axioms mathematics uses today are not arbitrary!! We picked them because they make mathematics match reality, and thus makes it useful. So in most cases, when ZFC says “X is true”, you can go do an experiment and find that X is true experimentally. For example, “0 =/= 1” (in the integers) is true in ZFC, but also true in reality because if I have 0 apples it’s not the same as having 1 apple. ZFC also makes some statements we can’t experimentally verify. For example, ZFC happily deals with notions of “infinities”, even though everything we’ve ever observed in reality is finite. Statements about infinities may be true in ZFC, but are unknown in reality. Now we get to the continuum hypothesis. As mentioned, ZFC simply can’t answer the question, and we can extend ZFC to make it either true or false. Your question about it being “really true” is asking _whether it’s true in reality._ However, the continuum hypothesis is a question about infinities, while we have only ever done experiments on finite things. Thus, we cannot answer any question about infinities in an experimental manner! If some day we find a way to do science on infinities - or find that infinities are useful for predicting the behaviour of finite objects which we can perform experiments on - we may eventually find that our universe either does or does not respect the continuum hypothesis. If that happens, we will extend mathematics with the relevant axiom required to make it match the universe. However, until that day comes there is no reason to prefer one answer over the other, so we just leave it at “we don’t know, and can’t know just yet”.
During my 2am wikipedia scrolling a while back I stumbled on Gödel's incompleteness theorems. I stopped reading at 4am still stuck on paragraph one. Thanks to you I understood 5% more of that paragraph. Now the question is, how do you prove that a set of axioms is consistent or worse w-conxistent. And also what does it mean to have recursively numerable theorems, and why is it important??? Brain exploding just reminiscing...
Glad I could make a 5% dent! The thing about consistency is that, if your theory is consistent, you won't be able to use your theory to prove its own consistency (unless it is a really weak theory). Therefore, we can only prove *relative* consistency results: "If theory A is consistent, then theory B is consistent." In this case, you prove theory B is consistent (relative to the consistency of theory A) by using theory A to build a *model* for theory B. Recursive enumerability of theorems is to say that you can write an algorithm down that generates all theorems of the theory (in the sense that any theorem of your theory will eventually be generated by the algorithm in finite time). Very roughly speaking, it means that there are "essentially finitely many" axioms and laws of deduction; you don't have an infinite collection of wildly different axioms that would make reasoning with the theory virtually impossible.
I find the use of a joke paralleling this funny but since I don't feel i have a full grasp. I feel I only half get the joke. But it's a great incentive to watch the video s few times to get it lol.
The main thing I’ve got from this is that you could make multiple models from the same set of axioms. But isn’t 2=0 another axiom, if you force it? Then it isn’t the same set of axioms… right?
The point is that the common way of proving some proposition P to be unprovable and undisprovable from a set of axioms is to provide two models for the axioms: one model where P is true, and another model where P is false. I didn't force "2 = 0" onto the axioms, but onto a *model* of the axioms (to build a new one). That being said, if you include "2 = 0" as an axiom to the list of axioms in our toy theory of integers, you do get a more specific theory: namely, the theory of "integral domains of characteristic 2", and it is true that there are several models for this theory as well. You can use these models to prove new statements to be unprovable/undisprovable. For example: the statement "x = 0 or x = 1" is unprovable and undisprovable in this theory. I hope this helps clarify some things!
Proving that something is false doesn't prove that the original statement is unprovable, because the system of axioms might be inconsistent (there may be some statement(s) that can be proven and disproven, or equivalently, it is possible to prove the negation of some axiom). For example, if I just have a simple set of axioms like: A) There exists an element 0. B) There exists an element 1. C) 0 ≠ 1 D) 0 = 1 Then I can prove that 0 ≠ 1 directly from axiom C, but this does not prove that I cannot prove the negation, that 0 = 1. And to demonstrate this, I can in fact prove that 0 = 1, directly from axiom D. Long story short, usually you need the caveat that says something like "if ZFC (or whatever axiom system you are talking about) is consistent, then...".
Yeah I was working with the hidden assumption that the ambient theory (in this case, the axioms of an integral domain) is consistent. Tom Prince made a similar remark in another comment iirc.
I didn't fully formalise the ambient system (for brevity). That a = b implies a + c = b + c can be chalked up to how addition defines a function (and thus preserves equality), or something like that. The axioms listed are built on implicit axioms for the primitives used.
how can you prove that a structure satisfies a given set of axioms? i'm a math undergrad very involved in logic, and this questions PLAGUES me, as i've searched litterally everywhere and i can't seem to find an answer, but everyone treats it as taken for granted (like lol obviously the integers satisfy these axioms, just look at them)
To prove that e.g. associativity of addition holds in the integers is to prove for all integers a, b, c that (ab)c = a(bc). This will boil down to how you decided to define integer addition. One way would be to define addition of natural numbers and then adjoin the negatives. In this case, addition is defined by induction, so you can prove its associativity in the natural numbers by induction. Passing to general integers can then be done with some simpler algebraic manipulations using the definition of negatives, accounting for whenever some of a, b, c are negative. Other axioms could be done similarly. Hopefully this helps if your aim is to verify the axioms of an integral domain for integers!
wait. when you talk about defining integers (and operations) based on the definition of the natural numbers and that being enough for the satisfaction of the axioms, that leads me to a confusion isn't the definition of the naturals literally the first-order theory N (the peano axioms)? if i naively extend your reasoning, i could prove that the naturals, as defined by the peano axioms, are OBVIOUSLY a model of N, proving it's unprovable consistency. i've been reading the Introduction to Metamathematics, and for the first time i've seen someone (kleene) explain the satisfiability consistency thing in a different way (describing the method before the hilbert program), he explained that a model is just another theory which can embed the original theory for which consistency is trying to be proven, e. g. euclidean geometry is consistent IF the theory for reals are consistent, but doesn't actually prove the consistency of euclidean geometry, and you could only actually prove the consistency by exhaustion if you could find a finite model. i didn't end the book so i don't know if anything has changed, what i know is that the godel kinda killed the program when proving the unprovability of the consistency of N (or any theory in which you could derive the naturals and arithmetic). For once, how could you prove that 0 belongs to the integers?
Right, I swept the detail of "where do these models come from??" under the rug (for the sake of keeping things simple). My model "the integers" needs to be constructed somehow, and to be precise, this means I'm building the integers inside of some ambient theory; for example, perhaps my ambient theory was ZFC. As a result, my implicit claims of consistency of the axioms of an integral domain (demonstrated by providing models of an integral domain) are only *relative* to the ambient theory wherein I'm building my models. Hope this helps!
Not every set of axioms has unprovable statements. For example, the system of integers from the video together with 2=0 is complete and consistent: every statement is either provable or disprovable. Gödel's incompleteness results state that every *sufficiently complex* axiom system is incomplete, and give a threshold for what "sufficiently complex" means. The theory of Z/2Z just isn't complex enough for the incompleteness results to apply.
There are at least infinitely many models satisfying ZFC (assuming it does not contain contradictions). Assume you have a valid model M with domain D that represents sets, and a relation IN that represents set membership “∈”. Now take any definable rule that uniquely ”assigns” every “set” from D to every “set” from D. This rule induced a bijective relation on D, which can be seen as a permutation. I will denote this idea as “F(a) = b”, where “a” and “b” are “sets” from D. It should be understood as “the rule F is satisfied for the ordered pair (a,b) and «b» is the only element that satisfies it if «a» is the first element” (the uniqueness follows from the definition of F). Some examples are: 1. F(a) = a, except F({})=F({ {} }) and F({ {} })={}. 2. F(a)=a+{ {} } if “{} not IN a”, F(a)=a-{ {} } if “{} IN a” Now consider a “permuted model” PerM with the same domain D that represents sets and a relation PerIN that represents set membership. The new membership rule is “a PerIN b” if and only if “a IN F(b)”. You can prove that all axioms of ZF are satisfied in this model, except possibly Foundation. For example, the empty set exists. Let’s call the representation of the empty set in D as Emp. Then by bijectivity of F, we know there is an element PerEmp with “F(PerEmp) = Emp”. Now the statement “a PerIN PerEmp” is satisfied iff “a IN F(PerEmp)” which is the same as “a IN Emp” which is always false. Therefore, “a PerIN PerEmp” is always false, which means “PerEmp” is the empty set of the new model. The other axioms are proven similarly. Edit: I meant ZF-
@@whynautchase I made a small correction to my original comment. These model ZF without C. Which means you can construct examples and counter-examples for the axiom of choice with them. I would call that a rather diverse set of models.
why do people have a philosophical problem with CH but not with Euclid’s fifth postulate(the parallel axiom)?? is there any difference between the two in the sense that both are provably unprovable and undisprovable?? 🤔
Models of geometry using different versions of the parallel axiom are all useful. So all of them are actually used. Concerning the axiom of choice, adding it to ZF is more fruitful than the contrary. The model of ZFC is used extensively. Up to now no new interesting result seem to have been found using the continuum axiom nor its negation. So it stays in limbo.
@@srghmaStrictly speaking, he posts whatever's interesting to him (at least I hope so), so at least that's covered Isn't that a bit of a personal thing to request though?
@@stefanalecu9532 I should not ask? humans will join brains and become one human being using Neuralink like in Nexus trilogy. Secrets are not possible in this universe. So, why not tell?
@@mt180extras The successor function is actually redundant. You can just add 1 to a. But if a = p-1 (mod p), where p is the characteristic, you get 0, so watch out for that.
@@biblebot3947 Yeah, but the successor only works for integer values. You can't iterate s x times (x being a real number, need not be natural), that's not how it works.
New Math: actually we deny the fact that objective, transcendental truths exist which underlie all logic, and that they are abstracted from the concrete, but thanks for these rules tho. Ontology as developed since Aristotle and thru Scholasticism: am I a joke to you?
This is all so dumb. The continuum hypothesis is obviously false because a subset of a set and a set are not the same size. So clearly they cannot be correlated. mapped onto each other. You can map all the numbers between 1 and 2 to all the numbers between 2 and 3, but if you map between 1 and 2 to between 2 and 4, the number is twice as large + 1 (the extra 1 being 3 itself.) Idk how math people get themselves stuck in such ridiculously obvious questions.
you can very easily map all the numbers between 1 and 2 to the numbers between 2 and 4. The map in question is just 2 times x. And the inverse is y divided by 2.
![[[OFFICIAL MV]] เอิ้นขวัญให้อ้าย- อ๊ะอาย สกุณาพร | ค่ายเพลงเป็นหนึ่ง](http://i.ytimg.com/vi/83x6L3X3qh0/mqdefault.jpg)
It doesn't detract from the video, but for the pedants, I'll point out that there are various points in the video where the logic depends on the hidden assumption that system of axions is consistent. In particular, the claim that a statement implying a contradiction means that the statement is unprovable depends on the system being consistent.
Wow! I've been interested in Godel's silly theorems for a while, and this was a great explanation of unprovability! Thank you!
Incompleteness*
Gödels silly theorems lol
when 0 deploys his integral domain expansion, every stranger loses it's cero-divisor qualities, even forcing them to have inverse in the finite domain
Is this a jjk reference that I don't get?
@@micayahritchie7158 yeah
7:34 Rings with characteristic 2: Allow us to introduce ourselves.
Oh this makes sense. What you're saying is that it is similar to the parallel postulate or ZFC? Where something is unprovable and not unprovable from a given set of axioms because it is possible to construct a system where it can work and where it doesn't from said axioms?
Yep, exactly!
I think the case of the parallel postulate is more interesting because it's more similar to actual ZFC modelling in a way
since you can, for example, directly construct spherical geometry FROM the geometric axioms, which doesn't satisfy the parallel postulate
I’m very happy see content like this. Please continue to make content.
🫡🫡🫡
This made things click in my brain! Multiple examples/realizations of a definition with different properties are extremely common in mathematics. There being different models of ZFC is only slightly weirder than that there are several different vector spaces. Gödel simply proved can NEVER pin things down to a single "true" model...
Gödel proved that there are unprovable true statements only for sufficiently powerful mathematical systems.
He used Peano Arithmetic for his proof, and his proof applies to any system that is ω-consistent.
Later extensions proved that it is true for any consistent system (which doesn't have much effect because almost all material systems that are consistent are also ω-consistent).
It is possible for simpler systems (like those that lack the concept of a "next number", or that have no reliable way to go from x to x+1) to be complete and consistent. Of course, such systems have extremely limited applicability.
these are actually good wtf thanks
A+ video. It took a while for me to understand the difference between "ZFC as foundations" and "ZFC as an object of mathematical study".
oh cool so that's what a model is 😃 thanks
The last line was great lmao. Your channel is a gem!
Just discovered your channel, giving you a well deserved subscriber!!
Only improvement i can recommend for the future is the audio quality 🙂
Great to hear that audio is the only major issue! (Exploring options to varying levels of success on that front)
best math channel in the game
this channel is gold, love the digestable videos, especially for someone like me with only an applied mathematics background (i barely know any pure math)
excellent and very accessible explanation of these concepts! as a bonus it also made it a lot clearer for me what the difference is between the single and double turnstiles, despite you not explicitly explaining them. thanks!
I just proved for odd q that (2^a(q-1/2) -1)/q that it is always an integer when a(x) is the sequence A002326 where a(0) is 1. This can be used to easily check Mersenne primes by setting it equal to 1. Sometimes it breaks like for M11 because multiple values of q work. To fix this, use the smallest value of 2^a(q-1/2) -1 for a chosen q. If it is 1, it is prime. If it isn’t, it is not prime.
I wanna thank you for popping up at the right time. As a physics student who's just completed my sophomore year, and started getting into math, this is both just advanced and simple enough for me to actually be able to learn from. Keep up the good work.
Thanks fam! Maybe with more nudging you'll switch programs 🙃
@@SheafificationOfG who knows lol. Definitely gonna teach myself some abstract algebra though. It's super interesting.
I'm currently taking an undergrad level logic course and this video is super helpful. Thank you!
Very practical demonstration.
2:36 There is a 3rd. Abstract algebra (specifically field theory/group theory).
What are you on about?
Middle dot (U+00B87) for multiplication? Everyone knows that the standard symbol for multiplication is Double curly loop (U+27BF)!
This really fibers my bundles, thank you.
My favourite part of a comment "section"
@@SheafificationOfG a mment cosection
Can't wait for your next videos. This one was excellent!
Very good video!
Thanks!
Wow this shattered my illusions of mathematics
You must have fucked up on this one because my dumb ass finally understood it
I like the noise. It's authentic.
If i summarize correctly, you make two models which are "subset" (?) of the model you want to use, then make them each prove a statement and its negation. This makes the statement unprovable and undisprovable.
If so, good job for the video!
Close, but these models aren't subsets of each other.
You have a theory, and a statement you're interested in.
To prove the statement is independent of the theory, you need two models of the theory (meaning all axioms of the theory hold in the model), where the statement is true in one model, and false in the other model.
@@SheafificationOfG "model of the theory" means the model is subset of the theory right? Or whatever the definition is where the model is the original theory + some additional axioms.
@@HazhMcMoor A theory is a set of rules, whereas a model of a theory is an actual thing that satisfies those rules.
The integers do not form a "subset" of the axioms of an integral domain.
This has been hinted at in another comment (I think?) but your demonstration at 8:56 only illustrates that our ring must have characteristic 2, not the much stronger claim that it's Z/2Z
Right, if you just take the axioms of an integral domain and tack on "2 =0" you get integral domains of characteristic 2. I didn't stress the point, but I mentiom prior to the construction that the idea behind "forcing" starts with picking an existing model. So, in my case, I was starting with the model Z (usual integers) and then asserting 2 = 0.
I was trying to mirror Cohen's approach to demonstrating the unprovability of CH: he started with a model where CH was true, and then used forcing to extend it to a model where CH failed.
great video! very informative
Very nice video! I just discovered your channel and can't wait to watch your other ones!
One thing I’m still a bit confused on is how this all plays with the law of excluded middle. If we take it as axiom, then these statements have truth value. Should be think of them being simultaneously true and false (until we create a model where we have chosen a specific value)?
The reasoning at 8:16 feels wrong - To me this sounds like “This statement is disprovable if we add this one rule, so if we remove this rule, we are no longer able to verify it”, like, if I want to prove if Goldbach Conjecture is unprovable, then I add a rule say that “Goldbach is false”, which it obviously is in that ruleset, and then I remove that rule, thus complete the proof that Goldbach is unprovable?
Thank you.
See pinned comment. You can add a rule "Goldbach is false", but then you also need your new set of rules to be consistent. And proving it is a hard task. With ring of modulo 2 it's easy - there are only 2 elements. You can check that this new rule never contradicts any previous ones, but I fear proving that your new rulesets with "Goldbach is true"/"Goldbach is false" rules are consistent would be as hard as actually trying to prove/disprove it.
Why did you use the one thinker statue that got blown up? Genuine question.
Also great vid, love your content from the few minutes I have seen of it :)
I.... didn't actually know that, just thought it was a picture of The Thinker!
Your pronunciation of "Gödel" is very good. I've never heard an american pronounce such a beautiful "ö".
Glad to hear I didn't butcher the name! (Btw, the only "American" I am is "North American" 🍁)
@@SheafificationOfG no you didn't at all. Oh I'm sorry, it is sometimes hard for me to recognize the difference between Canadian accents and US accents. I guess people are right in saying assumptions can make a fool out of you.
@@Diogenes_ofSinope Nah, don't worry about it; I don't think my accent is very distinct from the neighbours down south!
Interesting! Are there any other commonly used axiomatic systems (other than ZFC, I mean) of set theory where the continuum hypothesis *can* be proved (or disproved)? And what do we get, if we just add the trueness or falseness of the continuum hypothesis as yet another axiom to the existing ZFC axioms? We should get two different mathematical universes, right? Then maybe we could just explore both of them and use the one that we like better :-D ...I'm pretty sure, this must have been done already?
Taking a dip in the Xeeleeverse Logic Pool, eh?
There are some interesting consequences of CH or its negation (for example: if the continuum hypothesis is true, then there exists a function f(x, y) where the integrals f(x,y)dxdy and f(x,y)dydx over the unit square give different values; in general, the existence of such a function is independent of ZFC, but I'm not sure if it's equivalent to CH).
However, there hasn't been anything so earth-shattering that makes the mathematics community at large decide if CH is true or otherwise.
@@recursiveslacker7730 I didn't know about the "Xeeleeverse" - but looked it up. Looks interesting.
@@SheafificationOfG Wow - that's interesting! Can you point me to any resources where I can learn more about this fact? In that case, I think, I'd opt for the math universe where the CH is false. ...I kinda *want* these two integrals to give the same value - that somehow seems to be "intuitively" true to me. I can "see" it - we are computing the volume between the graph of f(x,y) and the xy-plane in both cases - that should not depend on how we compute it ....at least for continuous functions. For more erratic functions...like - say - some sort of 2D version of the Dirichlet function - hmm... I don't know if the integrals *should* still be equal...maybe not...dunno
@@MusicEngineeer Sorry, I lied to you: the existence of a function where the two iterated integrals over the unit square differ is not *equivalent* to CH, but it is a consequence of CH; I've edited the original comment.
You can get more information starting from en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC#Measure_theory and following references, I imagine. (The whole article has a bunch of independence results, some of which are tied to CH, so it's also an interesting read).
I enjoyed this and found it actually illuminating. Thanks.
Thanks for the timestamps and proper subtitle, the intro is so boring for this interesting topic at the title
I've usually taken unproveable and undisprovable to mean that we can't determine whether something is true, but is decidedly either trrue or false - we just can't prove that - but this video seems to suggest instead that the problem simply has no inherent truth or falsity at all. Am I understanding correctly?
In mathematics, “true” and “false” are relative to the axioms we work with. The continuum hypothesis is neither true nor false from ZFC, because its axioms simply do not force an answer. This means we can extend ZFC to make it either true or false. Basically, answering the question requires a new axiom, and valid axioms exist to give either answer.
In science, we consider something “true” if it correctly predicts the behaviour of the universe. This is what most people mean when they say “X is true”.
Technically, these two definitions of “truth” are _entirely independent!_ Mathematics is not formally obliged to match reality.
However, the axioms mathematics uses today are not arbitrary!! We picked them because they make mathematics match reality, and thus makes it useful. So in most cases, when ZFC says “X is true”, you can go do an experiment and find that X is true experimentally.
For example, “0 =/= 1” (in the integers) is true in ZFC, but also true in reality because if I have 0 apples it’s not the same as having 1 apple.
ZFC also makes some statements we can’t experimentally verify. For example, ZFC happily deals with notions of “infinities”, even though everything we’ve ever observed in reality is finite. Statements about infinities may be true in ZFC, but are unknown in reality.
Now we get to the continuum hypothesis. As mentioned, ZFC simply can’t answer the question, and we can extend ZFC to make it either true or false. Your question about it being “really true” is asking _whether it’s true in reality._ However, the continuum hypothesis is a question about infinities, while we have only ever done experiments on finite things. Thus, we cannot answer any question about infinities in an experimental manner!
If some day we find a way to do science on infinities - or find that infinities are useful for predicting the behaviour of finite objects which we can perform experiments on - we may eventually find that our universe either does or does not respect the continuum hypothesis. If that happens, we will extend mathematics with the relevant axiom required to make it match the universe. However, until that day comes there is no reason to prefer one answer over the other, so we just leave it at “we don’t know, and can’t know just yet”.
During my 2am wikipedia scrolling a while back I stumbled on Gödel's incompleteness theorems. I stopped reading at 4am still stuck on paragraph one. Thanks to you I understood 5% more of that paragraph. Now the question is, how do you prove that a set of axioms is consistent or worse w-conxistent. And also what does it mean to have recursively numerable theorems, and why is it important??? Brain exploding just reminiscing...
Glad I could make a 5% dent!
The thing about consistency is that, if your theory is consistent, you won't be able to use your theory to prove its own consistency (unless it is a really weak theory). Therefore, we can only prove *relative* consistency results: "If theory A is consistent, then theory B is consistent." In this case, you prove theory B is consistent (relative to the consistency of theory A) by using theory A to build a *model* for theory B.
Recursive enumerability of theorems is to say that you can write an algorithm down that generates all theorems of the theory (in the sense that any theorem of your theory will eventually be generated by the algorithm in finite time). Very roughly speaking, it means that there are "essentially finitely many" axioms and laws of deduction; you don't have an infinite collection of wildly different axioms that would make reasoning with the theory virtually impossible.
I find the use of a joke paralleling this funny but since I don't feel i have a full grasp. I feel I only half get the joke. But it's a great incentive to watch the video s few times to get it lol.
4:26 We need a video about 0-based indexing.
The main thing I’ve got from this is that you could make multiple models from the same set of axioms. But isn’t 2=0 another axiom, if you force it? Then it isn’t the same set of axioms… right?
The point is that the common way of proving some proposition P to be unprovable and undisprovable from a set of axioms is to provide two models for the axioms: one model where P is true, and another model where P is false.
I didn't force "2 = 0" onto the axioms, but onto a *model* of the axioms (to build a new one).
That being said, if you include "2 = 0" as an axiom to the list of axioms in our toy theory of integers, you do get a more specific theory: namely, the theory of "integral domains of characteristic 2", and it is true that there are several models for this theory as well.
You can use these models to prove new statements to be unprovable/undisprovable. For example: the statement "x = 0 or x = 1" is unprovable and undisprovable in this theory.
I hope this helps clarify some things!
@@SheafificationOfG yeah it does thanks bro
Proving that something is false doesn't prove that the original statement is unprovable, because the system of axioms might be inconsistent (there may be some statement(s) that can be proven and disproven, or equivalently, it is possible to prove the negation of some axiom).
For example, if I just have a simple set of axioms like:
A) There exists an element 0.
B) There exists an element 1.
C) 0 ≠ 1
D) 0 = 1
Then I can prove that 0 ≠ 1 directly from axiom C, but this does not prove that I cannot prove the negation, that 0 = 1. And to demonstrate this, I can in fact prove that 0 = 1, directly from axiom D.
Long story short, usually you need the caveat that says something like "if ZFC (or whatever axiom system you are talking about) is consistent, then...".
Yeah I was working with the hidden assumption that the ambient theory (in this case, the axioms of an integral domain) is consistent. Tom Prince made a similar remark in another comment iirc.
so good!!
4:54
you forgot the parenthesis on ((1+1)•(x•y)). rookie mistake.
DAMMIT
I don't see how 1 = 0 follows from 4 = 3, as you have not shown an inference rule from a = b => a + c = b + c. Or am I missing something?
I didn't fully formalise the ambient system (for brevity). That a = b implies a + c = b + c can be chalked up to how addition defines a function (and thus preserves equality), or something like that.
The axioms listed are built on implicit axioms for the primitives used.
Thanks
Awesome video
Came for the memes, stayed for the footnotes.
Nice video men !
Awesome video!
More set theory videos, please!
how can you prove that a structure satisfies a given set of axioms?
i'm a math undergrad very involved in logic, and this questions PLAGUES me, as i've searched litterally everywhere and i can't seem to find an answer, but everyone treats it as taken for granted (like lol obviously the integers satisfy these axioms, just look at them)
To prove that e.g. associativity of addition holds in the integers is to prove for all integers a, b, c that (ab)c = a(bc). This will boil down to how you decided to define integer addition.
One way would be to define addition of natural numbers and then adjoin the negatives. In this case, addition is defined by induction, so you can prove its associativity in the natural numbers by induction. Passing to general integers can then be done with some simpler algebraic manipulations using the definition of negatives, accounting for whenever some of a, b, c are negative.
Other axioms could be done similarly.
Hopefully this helps if your aim is to verify the axioms of an integral domain for integers!
@@SheafificationOfG ty so much
wait. when you talk about defining integers (and operations) based on the definition of the natural numbers and that being enough for the satisfaction of the axioms, that leads me to a confusion
isn't the definition of the naturals literally the first-order theory N (the peano axioms)? if i naively extend your reasoning, i could prove that the naturals, as defined by the peano axioms, are OBVIOUSLY a model of N, proving it's unprovable consistency.
i've been reading the Introduction to Metamathematics, and for the first time i've seen someone (kleene) explain the satisfiability consistency thing in a different way (describing the method before the hilbert program), he explained that a model is just another theory which can embed the original theory for which consistency is trying to be proven, e. g. euclidean geometry is consistent IF the theory for reals are consistent, but doesn't actually prove the consistency of euclidean geometry, and you could only actually prove the consistency by exhaustion if you could find a finite model. i didn't end the book so i don't know if anything has changed, what i know is that the godel kinda killed the program when proving the unprovability of the consistency of N (or any theory in which you could derive the naturals and arithmetic).
For once, how could you prove that 0 belongs to the integers?
Right, I swept the detail of "where do these models come from??" under the rug (for the sake of keeping things simple). My model "the integers" needs to be constructed somehow, and to be precise, this means I'm building the integers inside of some ambient theory; for example, perhaps my ambient theory was ZFC. As a result, my implicit claims of consistency of the axioms of an integral domain (demonstrated by providing models of an integral domain) are only *relative* to the ambient theory wherein I'm building my models.
Hope this helps!
If any set of axioms can make unprovably true statements, isn’t that saying for any set of axioms, you are missing one axiom?
Not every set of axioms has unprovable statements. For example, the system of integers from the video together with 2=0 is complete and consistent: every statement is either provable or disprovable.
Gödel's incompleteness results state that every *sufficiently complex* axiom system is incomplete, and give a threshold for what "sufficiently complex" means. The theory of Z/2Z just isn't complex enough for the incompleteness results to apply.
Good video!
nice video! -forcing explanation when-
Didn't expect to see you here
8:54 when in doubt always bet on Z/2Z
My GOAT
Wow I always thought that independence was a much stronger statement than that. So this means that there is more than one model satisfying ZFC?
There are at least infinitely many models satisfying ZFC (assuming it does not contain contradictions).
Assume you have a valid model M with domain D that represents sets, and a relation IN that represents set membership “∈”.
Now take any definable rule that uniquely ”assigns” every “set” from D to every “set” from D. This rule induced a bijective relation on D, which can be seen as a permutation.
I will denote this idea as “F(a) = b”, where “a” and “b” are “sets” from D. It should be understood as “the rule F is satisfied for the ordered pair (a,b) and «b» is the only element that satisfies it if «a» is the first element” (the uniqueness follows from the definition of F).
Some examples are:
1. F(a) = a, except F({})=F({ {} }) and F({ {} })={}.
2. F(a)=a+{ {} } if “{} not IN a”, F(a)=a-{ {} } if “{} IN a”
Now consider a “permuted model” PerM with the same domain D that represents sets and a relation PerIN that represents set membership.
The new membership rule is
“a PerIN b” if and only if “a IN F(b)”.
You can prove that all axioms of ZF are satisfied in this model, except possibly Foundation.
For example, the empty set exists. Let’s call the representation of the empty set in D as Emp. Then by bijectivity of F, we know there is an element PerEmp with “F(PerEmp) = Emp”.
Now the statement “a PerIN PerEmp” is satisfied iff “a IN F(PerEmp)” which is the same as “a IN Emp” which is always false.
Therefore, “a PerIN PerEmp” is always false, which means “PerEmp” is the empty set of the new model.
The other axioms are proven similarly.
Edit: I meant ZF-
@@fullfungo that's neat. Doesn't really feel like a different model in any substantial sense though
@@whynautchase I made a small correction to my original comment. These model ZF without C. Which means you can construct examples and counter-examples for the axiom of choice with them.
I would call that a rather diverse set of models.
that was a cool video i think
why do people have a philosophical problem with CH but not with Euclid’s fifth postulate(the parallel axiom)??
is there any difference between the two in the sense that both are provably unprovable and undisprovable?? 🤔
Models of geometry using different versions of the parallel axiom are all useful. So all of them are actually used. Concerning the axiom of choice, adding it to ZF is more fruitful than the contrary. The model of ZFC is used extensively. Up to now no new interesting result seem to have been found using the continuum axiom nor its negation. So it stays in limbo.
@@ahoj7720 was talking about the continuum hypothesis.
Superbe vidéo
Merci !
I love meme some mathematics.
You’re the goat
bigboxswe of math
0:58 Puh-LEASE. Grothendieck was so feeble, he couldn't accept an award, and Gödel was such a specialist it hurts. Literally not even being sarcastic.
WTF?
Can u make a video about Your worldview? (I just write randomly)
Not randomly, but "what is in my mind, what is interesting to me"
@@srghmaStrictly speaking, he posts whatever's interesting to him (at least I hope so), so at least that's covered
Isn't that a bit of a personal thing to request though?
@@stefanalecu9532 I should not ask?
humans will join brains and become one human being using Neuralink like in Nexus trilogy. Secrets are not possible in this universe. So, why not tell?
Title sounds like this is about God 😂
first
successor of first
@@JoaoVictor-xi7nh s(s(first))
@@mt180extras The successor function is actually redundant. You can just add 1 to a. But if a = p-1 (mod p), where p is the characteristic, you get 0, so watch out for that.
@@Gordy-io8sbin PA, addition is defined in terms of the successor
@@biblebot3947 Yeah, but the successor only works for integer values. You can't iterate s x times (x being a real number, need not be natural), that's not how it works.
New Math: actually we deny the fact that objective, transcendental truths exist which underlie all logic, and that they are abstracted from the concrete, but thanks for these rules tho.
Ontology as developed since Aristotle and thru Scholasticism: am I a joke to you?
Horrible audio, sloppy math and silly pictures.
No gracias. 🙅👋
I'd like to like this, but all the inane pictures make it hard to pay attention to what you're saying.
Fair enough! I'm still figuring out a good style for my videos.
This is all so dumb. The continuum hypothesis is obviously false because a subset of a set and a set are not the same size. So clearly they cannot be correlated. mapped onto each other.
You can map all the numbers between 1 and 2 to all the numbers between 2 and 3, but if you map between 1 and 2 to between 2 and 4, the number is twice as large + 1 (the extra 1 being 3 itself.)
Idk how math people get themselves stuck in such ridiculously obvious questions.
Bait ☝️😂
you can very easily map all the numbers between 1 and 2 to the numbers between 2 and 4. The map in question is just 2 times x. And the inverse is y divided by 2.