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 have found my new RUclips hole. Came to ground my understanding of Set Theory for future explorations, but staying for this fascinating and super long playlist.
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.
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.
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 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 have found my new RUclips hole. Came to ground my understanding of Set Theory for future explorations, but staying for this fascinating and super long playlist.
So beautifully well explained even to a non maths person.
Excellent synopsis.
This was an enlightening video. Amazingly presented
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.
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.
Excellent video. I really wish more teachers taught this way.
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 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.