Real Numbers as Dedekind's cuts, 12 Essence of Set Theory
ฝัง
- เผยแพร่เมื่อ 11 ต.ค. 2024
- Watch my videos Ad-free & free:
Odysee: odysee.com/@Mi...
Vimeo: vimeo.com/user...
Download: olsak.net/mirek...
Formal construction of the set of all the real numbers.
Download: www.olsak.net/m...
![[DRIP] ‘Love, Maybe’ PREVIEW](http://i.ytimg.com/vi/U7hwHp95m1k/mqdefault.jpg)
Great video. The fact that rationals have to be defined as equivalence classes and not literal sets or numbers is also something you encounter in abstract algebra when constructing field completions of integral domains. Z/nZ (modular arithmetic mod n) is also defined as equivalence classes, since it divides the integers into n parts.
The teacher actually has referred the equivalence class in Q.
"We just have to deal with a little complication, multiple fractions can correspond to the same rational number. For example, minus three over six should be the same number as two over minus four. But formally, the ordered pair (-3, 6) is different from the ordered pair (2, -4). A natural solution to this little complication could be an additional condition that we consider only fractions in a reduced form, that is a fraction with the smallest possible positive denominator." Such a good explanation!!!
Many many many people will watch this series lecture, this series is really great, you explained the most fundamental in math.
Thank you
Amazingly clear presentation, thank you so much
3:47 What about the constrction where (a, 0) = positive and (0, a) negative?
Nice trick!
Actually, some people even like to consider arbitrary pairs (a,b) of natural numbers with the meaning of "a-b", and taking a quotient already in the integer construction.
I _also_ meant why do we discard (0, 1), too; hence the timestamp
@@notmyname7698 Yes, I understood it as a trick to merge +0 and -0 automatically, without a need to discard anything.
"ADDITION MUST BE PROVEN BY A SPIRIT."
You are amazing!!! Wow!!! I'm surprised by you all time.
Why didn't you do complex numbers from set theory in this series?
Indeed, I could -- the simplest option is to just encode a+bi as a pair of real numbers (a,b). I didn't consider it too important to introduce complex numbers here, as I see them as a bit more "advanced" concept, and their set-theoretical construction as not particularly interesting.
Also, if you have seen my video on complex numbers, I like their geometrical introduction more. It is however more challenging to get that straight from set theory -- one would first need to introduce the theory of linear algebra and a dot product. Then, you can define complex numbers as linear mappings R^2 -> R^2 which (1) preserve the orientation (have non-negative determinant 1), (2) map every pair of orthogonal vectors to a pair of orthogonal vectors. Addition is point-wise addition, multiplication is function composition. (In fancy language, it is a specific subring of the ring of endomorphisms of R^2).
Another option is to introduce them through an algebraic theory. They are the algebraic completion of real numbers -- that doesn't specify the construction directly, rather saying that no matter which non-linear irreducible polynomial in real numbers you take, it will be a quadratic polynomial, and extending real numbers with a root of that polynomial gives an algebraically closed field isomorphic to complex numbers.
"DEVIL HEIST."
great video! thank you
thx
Great video!!!
Great visuals, but I really don’t understand much of your english.
Sorry, my English is not great (and especially was not when I was recording this). Do you know you can turn on the subtitles?
@@procdalsinazev Oh, I actually didn’t consider that. Thanks, I’ll rewatch it.
@@procdalsinazev note that I'm a native speaker and had no problem in understanding you
I understood him perfectly, although English is my native language so that might be a disclaimer
bro is 0blue4brown