I found the whole concept of countable & uncountable numbers amazing when I first came across them at uni. It's such a profound concept that 1) very few non mathematicians know about 2) is easy to show to non mathematicians
It is a very interesting topic. Very good explanation I remember that the subject struck me from the reading of Borges and Torretti's great book: "El paraíso de Cantor". David's English is as clear as Frank Sinatra´s English.
U are just awesome sir U explain things briefly but at greater level of depth Pls keep preparing videos on higher mathematics.. which are more abstract in nature
There is Aleph 0 (countably large set), then there is Aleph 1 (uncountably large set, such as real numbers on a number line, or set of all geometrical POINTS in space), then we have Aleph 2 (set of all geometrical CURVES in space, which has a density infinitely larger than the infinity of real numbers)
Actually, this is incorrect. The cardinality of the field of real numbers is not Aleph(1), it is 2^Aleph(0). Aleph(1) = 2^Aleph(0) IF AND ONLY IF the continuum hypothesis is true, but the continuum hypothesis cannot be proven from the standard axioms of set theory. Aleph(1) is the cardinality of the set of all countable ordinal numbers. The set of all geometrical curves in space has cardinality 2^(2^Aleph(0)), not Aleph(2).
What you’re thinking of is the Beth numbers, defined by ℶ₀ = ℵ₀ and ℶ_(n+1) = 2^(ℶ_n). It is actually undecidable from standard set theory whether or not ℶ₁ = ℵ₁. Also, believe it or not, the set of all geometrical curves in space actually still has cardinality ℶ₁, same as the set of real numbers! It actually takes quite a bit of work to find a set of cardinality ℶ₂; topology can get there (and of course set theory) but in almost all contexts we never get more than ℶ₁ elements.
@@TheBasikShow Ah, you are right. When I wrote the sentence about geometrical curves, I was actually thinking of the number of arbitrary functions from R^n to R^n, which is Beth(2). But geometric curves are necessarily continuous functions (except for maybe countably many exceptions), and so there can only be Beth(1) of them.
I enjoy maths just as much as a open enthusiast, teaching myself newer broader theories and hypothesizing my own. When I consider the Cardinality of infinite sets, would it not be the answer to those sets root back to 0, and 0 is a placeholder for the value of a number being 1...? Even if alif n were = 0, or having no values present, its given a value. One could hypothesize that the cardinality of all sets would be equal to 0 or at least an absolute value.
I found the whole concept of countable & uncountable numbers amazing when I first came across them at uni. It's such a profound concept that
1) very few non mathematicians know about
2) is easy to show to non mathematicians
It is a very interesting topic. Very good explanation I remember that the subject struck me from the reading of Borges and Torretti's great book: "El paraíso de Cantor". David's English is as clear as Frank Sinatra´s English.
Thank you!
Great video, many thanks.
many people can understand this video..I like the explanation 👍👍
Much more understandable than the other videos. ☺️
A very clear explanation. I'm looking forward to part 2.
Thanks very much. There are two separate videos to this called "Cardinals and Ordinals" - hope you enjoy them.
U are just awesome sir
U explain things briefly but at greater level of depth
Pls keep preparing videos on higher mathematics..
which are more abstract in nature
Thank you, Rajendra. More on their way.
extremely useful!
Thank you!
There is Aleph 0 (countably large set), then there is Aleph 1 (uncountably large set, such as real numbers on a number line, or set of all geometrical POINTS in space), then we have Aleph 2 (set of all geometrical CURVES in space, which has a density infinitely larger than the infinity of real numbers)
Indeed. We''ll be discussing the infinite cardinals (alephs) and ordinals (omegas, etc) in more detail in two upcoming videos.
Actually, this is incorrect. The cardinality of the field of real numbers is not Aleph(1), it is 2^Aleph(0). Aleph(1) = 2^Aleph(0) IF AND ONLY IF the continuum hypothesis is true, but the continuum hypothesis cannot be proven from the standard axioms of set theory. Aleph(1) is the cardinality of the set of all countable ordinal numbers. The set of all geometrical curves in space has cardinality 2^(2^Aleph(0)), not Aleph(2).
What you’re thinking of is the Beth numbers, defined by ℶ₀ = ℵ₀ and ℶ_(n+1) = 2^(ℶ_n). It is actually undecidable from standard set theory whether or not ℶ₁ = ℵ₁.
Also, believe it or not, the set of all geometrical curves in space actually still has cardinality ℶ₁, same as the set of real numbers! It actually takes quite a bit of work to find a set of cardinality ℶ₂; topology can get there (and of course set theory) but in almost all contexts we never get more than ℶ₁ elements.
@@TheBasikShow Ah, you are right. When I wrote the sentence about geometrical curves, I was actually thinking of the number of arbitrary functions from R^n to R^n, which is Beth(2). But geometric curves are necessarily continuous functions (except for maybe countably many exceptions), and so there can only be Beth(1) of them.
I love how I didn't understand anything and my teachers discussion about this
yes it really helps me out
Thanks 👏 btw is this lesson for 1st year high schoolers?
I enjoy maths just as much as a open enthusiast, teaching myself newer broader theories and hypothesizing my own.
When I consider the Cardinality of infinite sets, would it not be the answer to those sets root back to 0, and 0 is a placeholder for the value of a number being 1...? Even if alif n were = 0, or having no values present, its given a value. One could hypothesize that the cardinality of all sets would be equal to 0 or at least an absolute value.
I can't understand
There’s no way there are the same number of even numbers as there are even and odd numbers….
And yet it's true!