Mate I love your channel, it's criminally underrated. It's clear you have a real passion for mathematics. I graduated with a bachelors in Mathematics 4 years ago and I missed topics like these, we did a bit of set theory but never into the nitty-gritty like this. Would you ever consider doing a video of other constructions of the natural numbers like Peano axioms? Either way, love your content!
Love the video! I do have one small criticism wholly unrelated to the mathematics: the background music is too loud to the point that it gets a bit overwhelming when wearing headphones. Everything else was pretty much perfect.
As someone who is familiar with programming and have mild experience with Agda, this was delightful to watch! It seems like everything up to your definition of the integers is fairly standard in Agda, but the way you defined the integers isn’t. (It seems more directly related to how they are defined in cubical Agda, or at least 1Lab.)
Really nice overview of the motivation for being rigorous! Can you recommend a book or article to read up details? Especially about the step to the real numbers.
The construction from this video takes a lot from Elements of Set Theory by Herbet B. Enderton and Set Theory: An Open Introduction by Tim Button. The latter book is free to download here st.openlogicproject.org/.
at 32:52 , when presenting the dedekind cut for a rational, it's defined as `q < r`; does that imply that r itself isn't in the set defining Q? I feel it should be
@19:45 wouldn’t showing this is a transitive relation use subtraction? But that’s not an operation we have yet. We would have that (a,b)~(c,d)~(e,f) Then a+d=b+c c+f=d+e We want to show a+f=b+e So we can add the two equations to get a+d+c+f=b+c+d+e Then by canceling the d+c on both sides we get that a+f=b+e, hence (a,b)~(e,f). But the step of canceling the d+c term uses subtraction which we don’t have yet, so I’m unsure about that.
Similar question on showing transitive property for rationals. It’s obviously transitive using division, but again we are trying to define rationals, so the division operation is not yet defined to use it show this transitive property. That would be circular.
It turns out that this "cancellation" property of the naturals can be directly proven just by our natural number construction. The ability to subtract is really just the existence of additive inverses for all elements of a set. So while additive inverses imply cancellation, cancellation does not necessarily imply additive inverses (and thus subtraction). For the sake of time, I unfortunately had to skip these minor steps but the general idea is that we can prove the cancellation property in N by induction, and using the fact that the natural numbers are what's called "transitive sets".
@@jHan thanks, I appreciate this clarification. I think I’ve seen this proof before, but I thought the video was saying that the transitive step is trivial, but I think there’s something to show there using the induction. Quite nice that transitive property of the naturals gives us the cancellation rule which shows the transitive relation here.
I liked this presentation, but I feel like the last 30 seconds suddenly brought up some controversial claims with no backing. "Set theory isn't a painstaking formalization for the sake of formalization" I feel like that was why Set Theory was created, though that's certainly not what modern Set theorists do. "The axioms, theorems, and tools of Set theory have applications in every branch of mathematics" I feel like, with the possible exception of the Axiom of Choice, the axioms intentionally don't have applications. They're just a foundation from which we can get things the rest of math needs, like the fundamental property of the ordered pair. And the theorems and tools of Set theory, like forcing and the results that come from it, have essentially no bearing on other fields of mathematics. To be clear, I enjoyed my graduate Set theory classes and I like formalization! I just don't see things the same way.
jHan is awesome. And handsome. But "is zero a natural number?"? Man ... the answer to that is very simple: each person decides ... it is a matter of convention. If you want to consider zero a natural number, ok. If you don't consider zero a natural, that's also ok. THAT'S the answer. There is nothing more to it. I don't understand why people still ask such question ...
great, to say there's nothing is to be god. in the beginning there is nothing, and then god create 0, there there is nothing and 0. and from that all math and universe created. hahaha
