One of the best introductions to Naive Set Theory on the internet. You covered the history, all the essential points and then some in this video. A real gem, honestly.
This is really interesting although I had to rerun the proof of Cantor's Theorem a few times. Two things I took from this. 1. Why naive set theory is called naive set theory. 2. I look after a database of crashes. We used to have 26 categories of object struck by a vehicle. When a new system was developed someone thought that we should have subcategories of these categories. After two years we decided that we didn't have enough subcategories. We tried to rationalise the categories and subcategories but this has proven too difficult. I realised that this is a job for category theory. This video shows why category theory is tough.
Unrestricted comprehension is problematic because it permits the introduction of concepts/properties/elements that are illusory. Russell proposed that the set of all sets that do not refer to themselves (call this the comprehensive set) must be a member of itself. It can be easily shown (using classical logic) that this is incorrect, even with unrestricted comprehension, because adding such set to itself has no substantive effect on (i.e., does not substantively change the makeup of) the comprehensive set. In other words, the elements of the comprehensive set are substantively identical both before and after adding such set to itself. Consequently, to add this set to itself is to require performance of a formal and unnecessary formal act that has no substance. And to give substantive effect to a non-substantive act is to elevate form over substance. I have written a paper that provides the detail if you are interested.
That's handwaving. Two sets are equal, if they have the same elements (mathematically: A=B, if and only if ∀x: x∈A if and only if x∈B). So the universal set U (set of all sets), and the same set with U itself removed (U'={x: x≠U}), are different sets. Naive set theory implicitly assumed that for every proposition P(x) there exists a set {x: P(x) is true}; more formally: ∃C: ∀x: x∈C if and only if P(x) is true. But this assumption turned to be inconsistent - let the proposition be "x∉x"; it can be seen that the set {x: x∉x} cannot exist. Let D be any set. There are exactly two possibilities: either D∉D, or D∈D, and in either case D doesn't have the property that x∈D if and only if x∉x. (The property is contradicted precisely by the set D itself; just like if a male barber claims to shave all men in his town who don't shave themselves, and nobody else, he can't be telling the truth - if you ask him: "do you shave yourself?", he can't answer the question without contradicting the previous claim.)
Understanding mathematics was once believed to be a human construct, and that being said one must consider theoretical naive set theory. That being a set or collection of number series sets. Theoretical naive set theory is a conglomerate of complexity utilizing infinity sets. It may be logical to presume, yet illogical to conclude, paradoxical in nature and finite in application.
Great vid! I'm currently working on set theory problem from Charles Pinter book where I'm asked to show that a Russell class {x: x doesn't belong to x} is a proper class, hence not a set. My intuition tells me to prove this by contradiction, assuming that Russel class is a set belonging to some class A... How to proceed to get a contradiction ?
One of the best introductions to Naive Set Theory on the internet. You covered the history, all the essential points and then some in this video. A real gem, honestly.
This is really interesting although I had to rerun the proof of Cantor's Theorem a few times. Two things I took from this. 1. Why naive set theory is called naive set theory. 2. I look after a database of crashes. We used to have 26 categories of object struck by a vehicle. When a new system was developed someone thought that we should have subcategories of these categories. After two years we decided that we didn't have enough subcategories. We tried to rationalise the categories and subcategories but this has proven too difficult. I realised that this is a job for category theory. This video shows why category theory is tough.
With this video, I am finally able to learn the true essence of Russell's Paradox and where did it come from.
Unrestricted comprehension is problematic because it permits the introduction of concepts/properties/elements that are illusory. Russell proposed that the set of all sets that do not refer to themselves (call this the comprehensive set) must be a member of itself. It can be easily shown (using classical logic) that this is incorrect, even with unrestricted comprehension, because adding such set to itself has no substantive effect on (i.e., does not substantively change the makeup of) the comprehensive set. In other words, the elements of the comprehensive set are substantively identical both before and after adding such set to itself. Consequently, to add this set to itself is to require performance of a formal and unnecessary formal act that has no substance. And to give substantive effect to a non-substantive act is to elevate form over substance. I have written a paper that provides the detail if you are interested.
That's handwaving. Two sets are equal, if they have the same elements (mathematically: A=B, if and only if ∀x: x∈A if and only if x∈B). So the universal set U (set of all sets), and the same set with U itself removed (U'={x: x≠U}), are different sets. Naive set theory implicitly assumed that for every proposition P(x) there exists a set {x: P(x) is true}; more formally: ∃C: ∀x: x∈C if and only if P(x) is true. But this assumption turned to be inconsistent - let the proposition be "x∉x"; it can be seen that the set {x: x∉x} cannot exist. Let D be any set. There are exactly two possibilities: either D∉D, or D∈D, and in either case D doesn't have the property that x∈D if and only if x∉x. (The property is contradicted precisely by the set D itself; just like if a male barber claims to shave all men in his town who don't shave themselves, and nobody else, he can't be telling the truth - if you ask him: "do you shave yourself?", he can't answer the question without contradicting the previous claim.)
Terrence Howard
Understanding mathematics was once believed to be a human construct, and that being said one must consider theoretical naive set theory. That being a set or collection of number series sets. Theoretical naive set theory is a conglomerate of complexity utilizing infinity sets.
It may be logical to presume, yet illogical to conclude, paradoxical in nature and finite in application.
terrence howard
Great vid! I'm currently working on set theory problem from Charles Pinter book where I'm asked to show that a Russell class {x: x doesn't belong to x} is a proper class, hence not a set.
My intuition tells me to prove this by contradiction, assuming that Russel class is a set belonging to some class A... How to proceed to get a contradiction ?
Really helpful. Thank you
On a similar note can you do a video about Wttenstein's Tractatus Logico-Philosophicus or Gödels incompleteness theorem?
Greetings, this was an imformative video. is it okay to ask for the powerpoint presentation? Thank you
Helpful, thank you.
super helpful - ty
thanks