I have found my new TH-cam hole. Came to ground my understanding of Set Theory for future explorations, but staying for this fascinating and super long playlist.
I know I’m four years late to this video, but I’m so thankful to have found it. You’re a wonderful communicator and the subject matter is profoundly interesting. I’m so glad you produce content. Will be hunting you down on Patreon.
I agree that we shouldn't feel obligated to do all math inside the framework of Set Theory. However, it's important to preface that the basic notion of a set is inescapable (you can't think without a notion of sets). For that reason, I see why so many people want to start with Set Theory. In my view, two things can be true at once: sets are foundational, but we don't have to construct all of mathematics from sets.
You can very well think without a notion of a set, you can instead think with properties and relations. For example, instead of of thinking of the set of real numbers, you can think of the property of being a real number, or the predicate "is a real number". These concepts are available informally in natural language and formally in standard predicate logic. Predicate logic doesn't require set theory, but set theory requires predicate logic to even state the axioms of set theory. Moreover, classical predicate logic is very natural, since it captures the logic of natural language arguments. Set theory is far less natural, it requires many non-trivial assumptions related to infinity. The fact that we ended up with ZFC as the default set theory is probably mainly a historical artifact, since there are other available axiomatizations of set theory that seem similarly plausible. In contrast, there is really just one notion of first-order or higher-order predicate logic that that is simple and obvious, it's just called classical logic.
@@cube2fox I'm fully aware of the fact that you can construct Set Theory inside a logical framework. However, in order to study logic, we require a meta-theory, which will always involve the intuitive notion of a set and counting. In fact, the definition of an axiomatic system is: "a finite set (collection, body, list, sequence, or whatever you choose to call it) of symbols, axioms, and rules of inference." This is my point. The intuition of a set is inescapable, not Set Theory. Propositional logic cannot have absolute precedence over sets and counting, and vice versa. I found a similar discussion: math.stackexchange.com/questions/173735/how-to-avoid-perceived-circularity-when-defining-a-formal-language
@@EthnHDmlle The meta theory can be expressed in natural language without ever mentioning sets. Like, instead of talking about "the set of inference rules" you can simply talk about "the inference rules".
@@EthnHDmlle You simply don't need to assume them. It's like saying "just because you choose not zo acknowledge God, that doesn't mean He is not there." Which is true but doesn't show that He is there, or a necessary assumption.
Philosophically, the concept of "infinity" is simply an aspect of what the human animal cognizes as "Unknown." It cannot be quantified, and it cannot be measured.
Holy hell. Thanks for this. Lines up with my intuitions. I have been looking for this information for years. Every set theory video explains what sets are and never why they are. The bit about set theory needing axioms - which if I am understanding right - completely undermines the main rationale for the whole set theory project in the first place.
Excellent summary. Its interesting how they detached mathematics from philosophy (calling it illegitimate to them) just long enough to starve mathematics of monotheists and breathe in misdirecting chaos into the natrative before gluing it back into philosophy. I'm working on a theory I call neural relativity, and the foundations of that I call dialectic topology. The foundation becomes the neural mesh, and differential reasoning becomes neural Interferometry. How mathematical ideas evolve is well accounted for by population interactions of neurally relative value judgements.
I am by no means an expert on the subject, but I seem to understand that in someway “there is no other way” than using something “like“ set theory to define mathematics. However, I don’t really know why. I think that this is because Russell’s attempt at building from the ground up using Peano axioms failed, in large parts due to the incompleteness theorem from Kurt Gödel. Though to be frank I am not too sure if that’s exactly the story. However, it is true that I am never entirely convinced whenever I read a proof in that field. One striking example is that I find it really hard to agree that if a set contains the thing that is nothing i.e the empty set then that set is not nothing, ie not empty. Usually when you reach that point they say something like “it takes convincing oneself but it works”. So if that statement was added as an axiom I would be more comfortable actually.
To be frank I can't see a single thing your change in foundation solves for mathematics. I have heard many mathematical philosophers express a desire to find new axioms to expand math or solve issues for example the famous joke about Choice, Zorn's and well ordering. You are though the fisrt I have heard who wants to discard infinity itself. I think you need to clarify in this video actual problems rather than gesture at what seems to amount to "bad vibes" you get from the reals and infinity. You say this is where the difficulties begin but you haven't named them. I am genuinely enthusiastic for someone to attempt to reconstruct modern mathematics from a finitist point of view but you still have all your work set out for you to say that this is necessary. Modern mathematics has grown corn so to speak (one you may dislike perhaps), you're vision still needs to prove that it can. Lastly, I have a philosophical issue with your framing of axioms as somehow inherently bad. It's my perspective that all mathematics must necessarily come down to an axiomatic sense, you can hide it behind intuition but that in my opinion is just a sloppy handing off of them and not an actual solution.
Well there is a simple problem: that so-called “real number arithmetic” can’t actually produce exact correct answers. Is that sufficiently bothersome for you? To be specific here is my standard chestnut : compute pi+e+sqrt(2). I claim that if pure mathematics cannot solve this simple arithmetical problem without chicanery, then our arithmetic is a logical sham.
@njwildberger Is the chicanery here the convergence of infinite series? Or using partial sums to compute approximations of these? None of this to me is even close to problematic because it provides sufficiently useful and self consistently logical answers. Please try again to find an actual "failure". To me a proper metamathematical view on whether our system serves us is if it gives us the broadest self consistent system that supports pragmatic discoveries and more. Perhaps we could do better, but you can't deny current math fulfills this.
@@njwildberger It's approximately 7.274088..., it can be computed to an arbitrary error using infinite series or other tools. It is true I cannot write down it's "exact" value in decimals, but the approximations get as close as you could ever need them to be. Also writing down it's expression as an infinite sum is in some sense relating it's exact value in another form of language. I see no issue with this conceptually or practically at all. No weakness in logic or "chicanery". Just ingenious methods developed by brilliant people.
Just recently found your videos after getting interested from a Russell/logic/philosophical angle. I must say, your ability to explain mathematics and concepts to a non mathematician is excellent. I’m really enjoying all the foundational/logic videos. I’m excited to venture deeper into all mathematics.
greetings... since we cant understand the creation of prime numbers, all our perception of set theories will remain short of accuracy , the acuuracy the univers is build with.
I have found my new TH-cam hole. Came to ground my understanding of Set Theory for future explorations, but staying for this fascinating and super long playlist.
I know I’m four years late to this video, but I’m so thankful to have found it.
You’re a wonderful communicator and the subject matter is profoundly interesting. I’m so glad you produce content. Will be hunting you down on Patreon.
I agree that we shouldn't feel obligated to do all math inside the framework of Set Theory. However, it's important to preface that the basic notion of a set is inescapable (you can't think without a notion of sets). For that reason, I see why so many people want to start with Set Theory. In my view, two things can be true at once: sets are foundational, but we don't have to construct all of mathematics from sets.
You can very well think without a notion of a set, you can instead think with properties and relations. For example, instead of of thinking of the set of real numbers, you can think of the property of being a real number, or the predicate "is a real number".
These concepts are available informally in natural language and formally in standard predicate logic. Predicate logic doesn't require set theory, but set theory requires predicate logic to even state the axioms of set theory.
Moreover, classical predicate logic is very natural, since it captures the logic of natural language arguments. Set theory is far less natural, it requires many non-trivial assumptions related to infinity. The fact that we ended up with ZFC as the default set theory is probably mainly a historical artifact, since there are other available axiomatizations of set theory that seem similarly plausible.
In contrast, there is really just one notion of first-order or higher-order predicate logic that that is simple and obvious, it's just called classical logic.
@@cube2fox I'm fully aware of the fact that you can construct Set Theory inside a logical framework. However, in order to study logic, we require a meta-theory, which will always involve the intuitive notion of a set and counting. In fact, the definition of an axiomatic system is: "a finite set (collection, body, list, sequence, or whatever you choose to call it) of symbols, axioms, and rules of inference."
This is my point. The intuition of a set is inescapable, not Set Theory. Propositional logic cannot have absolute precedence over sets and counting, and vice versa.
I found a similar discussion:
math.stackexchange.com/questions/173735/how-to-avoid-perceived-circularity-when-defining-a-formal-language
@@EthnHDmlle The meta theory can be expressed in natural language without ever mentioning sets. Like, instead of talking about "the set of inference rules" you can simply talk about "the inference rules".
@@cube2fox Just because you choose not to acknowledge it, that doesn't mean it's not there. A set, in it's most basic form, is incredibly general.
@@EthnHDmlle You simply don't need to assume them. It's like saying "just because you choose not zo acknowledge God, that doesn't mean He is not there." Which is true but doesn't show that He is there, or a necessary assumption.
So beautifully well explained even to a non maths person.
Philosophically, the concept of "infinity" is simply an aspect of what the human animal cognizes as "Unknown." It cannot be quantified, and it cannot be measured.
Holy hell. Thanks for this. Lines up with my intuitions. I have been looking for this information for years. Every set theory video explains what sets are and never why they are. The bit about set theory needing axioms - which if I am understanding right - completely undermines the main rationale for the whole set theory project in the first place.
Excellent summary.
Its interesting how they detached mathematics from philosophy (calling it illegitimate to them) just long enough to starve mathematics of monotheists and breathe in misdirecting chaos into the natrative before gluing it back into philosophy.
I'm working on a theory I call neural relativity, and the foundations of that I call dialectic topology. The foundation becomes the neural mesh, and differential reasoning becomes neural Interferometry. How mathematical ideas evolve is well accounted for by population interactions of neurally relative value judgements.
I am by no means an expert on the subject, but I seem to understand that in someway “there is no other way” than using something “like“ set theory to define mathematics. However, I don’t really know why. I think that this is because Russell’s attempt at building from the ground up using Peano axioms failed, in large parts due to the incompleteness theorem from Kurt Gödel. Though to be frank I am not too sure if that’s exactly the story. However, it is true that I am never entirely convinced whenever I read a proof in that field. One striking example is that I find it really hard to agree that if a set contains the thing that is nothing i.e the empty set then that set is not nothing, ie not empty. Usually when you reach that point they say something like “it takes convincing oneself but it works”. So if that statement was added as an axiom I would be more comfortable actually.
The reason you think that is simple: it's called indoctrination.
To be frank I can't see a single thing your change in foundation solves for mathematics. I have heard many mathematical philosophers express a desire to find new axioms to expand math or solve issues for example the famous joke about Choice, Zorn's and well ordering. You are though the fisrt I have heard who wants to discard infinity itself. I think you need to clarify in this video actual problems rather than gesture at what seems to amount to "bad vibes" you get from the reals and infinity. You say this is where the difficulties begin but you haven't named them. I am genuinely enthusiastic for someone to attempt to reconstruct modern mathematics from a finitist point of view but you still have all your work set out for you to say that this is necessary. Modern mathematics has grown corn so to speak (one you may dislike perhaps), you're vision still needs to prove that it can.
Lastly, I have a philosophical issue with your framing of axioms as somehow inherently bad. It's my perspective that all mathematics must necessarily come down to an axiomatic sense, you can hide it behind intuition but that in my opinion is just a sloppy handing off of them and not an actual solution.
Well there is a simple problem: that so-called “real number arithmetic” can’t actually produce exact correct answers. Is that sufficiently bothersome for you?
To be specific here is my standard chestnut : compute pi+e+sqrt(2).
I claim that if pure mathematics cannot solve this simple arithmetical problem without chicanery, then our arithmetic is a logical sham.
@njwildberger Is the chicanery here the convergence of infinite series? Or using partial sums to compute approximations of these? None of this to me is even close to problematic because it provides sufficiently useful and self consistently logical answers. Please try again to find an actual "failure". To me a proper metamathematical view on whether our system serves us is if it gives us the broadest self consistent system that supports pragmatic discoveries and more. Perhaps we could do better, but you can't deny current math fulfills this.
@@itsamefkjionn6803 I just gave you a specific challenge. Why did you not even mention it in your reply?
@@njwildberger It's approximately 7.274088..., it can be computed to an arbitrary error using infinite series or other tools. It is true I cannot write down it's "exact" value in decimals, but the approximations get as close as you could ever need them to be. Also writing down it's expression as an infinite sum is in some sense relating it's exact value in another form of language. I see no issue with this conceptually or practically at all. No weakness in logic or "chicanery". Just ingenious methods developed by brilliant people.
@@njwildbergerplease write down 5!!!!!!!!!!!!!!!!. It’s a finite natural number. I want every digit, thanks.
Excellent synopsis.
This was an enlightening video. Amazingly presented
Just recently found your videos after getting interested from a Russell/logic/philosophical angle. I must say, your ability to explain mathematics and concepts to a non mathematician is excellent. I’m really enjoying all the foundational/logic videos. I’m excited to venture deeper into all mathematics.
a pleasant tour
greetings... since we cant understand the creation of prime numbers, all our perception of set theories will remain short of accuracy , the acuuracy the univers is build with.
Excellent video. I really wish more teachers taught this way.
The set containing all other sets seems to remain a bit of a problem, oh well, just ignore the impossibility of it! Simples... (:-)
To the contrary, we deploy a system, ZFC, in which that set does not exist.