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Infinite_1 sounds weird, but I guess aleph_0 does too. Also, if you want to trigger math students and former math students, sometime, you should leave a "left as exercise for the viewer" that's unanswered, but it crucial to understanding the rest. You probably aren't as much of a troll as I am, though.
Thanks for your comment! I decided to go without introducing aleph numbers within this video. Mostly because I hate the symbol. It's worth noting that the cardinal numbers I call infinity_n correspond exactly with the numbers beth_(n-1). They only correspond with aleph_(n-1) if we adopt the generalized continuum hypothesis, which is one of the additional axioms I was thinking about in my closing remark.
@@EpicMathTime Doesn't it still work for infinity_1 and aleph_0 though? I know that the generalized continuum hypothesis is needed elsewhere, but I thought aleph_0 was supposed to be the cardinality of the natural numbers. Been a while, though.
@@zero132132 Yes that's correct, aleph_0 is the cardinality of the natural numbers, as is beth_0. The aleph numbers are literally "the infinite cardinals, whatever the hell those are." Aleph_0 is certainly the cardinality of the naturals, but we can't say anything whatsoever about aleph_1 besides that it is "the next infinite cardinal after aleph_0." Once we adopt the continuum hypothesis, we can say that aleph_1 is the cardinality of R. When we adopt the generalized continuum hypothesis, we can say that the beth numbers and aleph numbers correspond. I think the beth numbers are a great pedogogical tool, if nothing else, for an introduction into infinite cardinals, because everything presented in this video is true within ZFC. Statements that are independent of ZFC (like the continuum hypothesis) are something I find extraordinarily interesting, though. I'm not sure which arena I prefer, because a set-theoretic universe in which every infinite cardinal can be identified with a power set is nice and structural, but on the other-hand seems limited.
Aleph(1) is defined as the cardinality of the set of all countable sets. Aleph(2) is defined as the cardinality of the set of all sets with cardinality Aleph(1). And so on. So the Aleph numbers have a recursive definition too, but this definition is not an intuitive or as easy to explain as with the Beth numbers, and there is also very little you can learn about the Aleph numbers at first glance from their definition alone.
8:34 wait, I just thought of something. P(N) also has countably infinite amount of elements. Why does it say here that it's the second type, the uncountable one? Or is there something I'm missing. Why it's countable is easily proven. There exists a function that, for any element of P(N), it makes a sum of powers of 2 such that the exponents of those powers are the elements of that element. It is verifiable that that function is a bijection, therefore |P(N)| is a countable infinity as well
@@EpicMathTime I put it in my comment but basically we can count them. Ø, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3} etc etc. I thought about what I said and maybe this counting doesn't work since we cannot count all the elements with infinite elements. Maybe that is a mistake in my hypothesis?
@@fountainovaphilosopher8112 Yeah, that's a big issue. I'm not sure, but I do believe the set of /finite/ subsets of natural numbers is countable. Consider that each real number can be represented as a decimal, which can be identified with a sequence of natural numbers.
@@EpicMathTime yes, the number of finite sets is certainly countable. Ok, thank you. Now I see better and agree that it's uncountable. I can just apply the similar logic that is applied to prove real numbers are uncountable.
@@Quasarbooster You're welcome. What gets tricky is when you consider "the set of all sets", which seems innocent enough, but isn't. Short answer, this leads to things like Russell's Paradox and proper classes.
@@Quasarbooster I've come across a book on category theory which went one step beyond proper classes to "illegitimate conglomerates". In fact, I think there's a set theory which goes omega levels beyond proper classes.
Well, they still exist in other ways, just like how "imaginary" numbers also exist about just as much. These are just names, and ultimately, they don't mean anything. The names that they have in many other languages probably don't even translate to this. It's more important to be able to understand how the respective sets the names refer to are defined and what properties they have.
Basically, could you say x^(InfinityN) =Infinity(N+1) if x>1 by a finite amount? Cause when you take then cordinality of the real, it's kinda like 10^(infinite1) and the power set of the natural is kinda like 2^(infinte1)... ?
True! The set of all real functions has larger cardinality, but with the restriction of continuity, we have the cardinality of the reals. A helpful fact to show that is that, since the rationals are dense in a reals, a continuous real function is completely determined by its value at the rational numbers.
I didn't take the continuum hypothesis as granted. The (extended) continuum hypothesis just puts the aleph numbers and the beth numbers in agreement, but the continuum hypothesis isn't needed for the beth numbers to just... exist.
@@EpicMathTime You are right. Anyway, I think you could have mentioned it so profanes don't think the beths and the alphas are necessarily the same. That could make a good extra video.
I think you're right, a video on the continuum hypothesis would be awesome. I do regret adopting this non-standard "infinity-1" convention back when I made this video (it seemed like a good idea at the time). But yeah, these cardinals "infinity-1" correspond with the beth numbers, not the aleph numbers (without CH), and my nonstandard notation really obscured this detail. Also, this same topic was discussed in replies to the pinned comment if you want to check that out.
It would be interesting to tell us more about those axioms required for aleph 4, 5... (or infinity 4, 5...). I am fascinated by axioms in math because, we rely on them without any way of proving or disproving them.
@@EpicMathTime it is a theorem of ZF that there is no set of cardinality "between" aleph 0 and aleph 1. First, it is well-known that a) every well-orderable set has an aleph number by definition (as the least order type), and that b) there is no aleph number between aleph 0 and aleph 1 by well-ordering of ordinals. Let A be a set. If |A| is less than or equal to aleph 1, then there is an injection between A and a set of cardinality aleph 1, meaning A is well-orderable, and thus |A| (being an aleph number) cannot be between aleph 0 and aleph 1. (Edit 2:) The messed-up thing about ZF without Choice is that there may be sets whose cardinalities aren't even comparable (this was proven to be equivalent to AC). Edit: It is also a theorem of ZF (immediately apparent from the von Neumann ordinal definition) that aleph 1 is the cardinality of the set of all ways to well-order the naturals. The Continuum Hypothesis states that this set of well-orderings on N has the same cardinality as that of P(N), or the real numbers. With the Axiom of Choice in hand, since every set is well-orderable and thus has an aleph number, we can say succinctly that the Continuum Hypothesis states there is no cardinality between N and R, i.e., R is of cardinality aleph 1.
I found it enlightening when a professor pointed out that how we think about numbers affects the cardinality of sets of numbers. If 3 is just an object "3" then a set {3} has cardinality one. If 3 is a natural number as defined by ZF-set theory then the cardinality of {3} is 3, since 3 := {{{}}} u {{}} u {} If 3 is taken as a real number it's usually defined as the limit of cauchy sequences of rationals (the other definitions are equivalent) and the set of all those sequences are bijective to 2^ *Q* so the cardinality of {3} is the same as the cardinality of reals so card{3} = card{ *R* }
Infinite_1 sounds weird, but I guess aleph_0 does too. Also, if you want to trigger math students and former math students, sometime, you should leave a "left as exercise for the viewer" that's unanswered, but it crucial to understanding the rest. You probably aren't as much of a troll as I am, though.
Thanks for your comment! I decided to go without introducing aleph numbers within this video. Mostly because I hate the symbol.
It's worth noting that the cardinal numbers I call infinity_n correspond exactly with the numbers beth_(n-1). They only correspond with aleph_(n-1) if we adopt the generalized continuum hypothesis, which is one of the additional axioms I was thinking about in my closing remark.
@@EpicMathTime Doesn't it still work for infinity_1 and aleph_0 though? I know that the generalized continuum hypothesis is needed elsewhere, but I thought aleph_0 was supposed to be the cardinality of the natural numbers. Been a while, though.
@@zero132132 Yes that's correct, aleph_0 is the cardinality of the natural numbers, as is beth_0.
The aleph numbers are literally "the infinite cardinals, whatever the hell those are." Aleph_0 is certainly the cardinality of the naturals, but we can't say anything whatsoever about aleph_1 besides that it is "the next infinite cardinal after aleph_0."
Once we adopt the continuum hypothesis, we can say that aleph_1 is the cardinality of R. When we adopt the generalized continuum hypothesis, we can say that the beth numbers and aleph numbers correspond.
I think the beth numbers are a great pedogogical tool, if nothing else, for an introduction into infinite cardinals, because everything presented in this video is true within ZFC.
Statements that are independent of ZFC (like the continuum hypothesis) are something I find extraordinarily interesting, though. I'm not sure which arena I prefer, because a set-theoretic universe in which every infinite cardinal can be identified with a power set is nice and structural, but on the other-hand seems limited.
Aleph(1) is defined as the cardinality of the set of all countable sets. Aleph(2) is defined as the cardinality of the set of all sets with cardinality Aleph(1). And so on. So the Aleph numbers have a recursive definition too, but this definition is not an intuitive or as easy to explain as with the Beth numbers, and there is also very little you can learn about the Aleph numbers at first glance from their definition alone.
Let's talk about sets baby.
Let's talk about A and B.
Let's talk about all the functions, from the Naturals to the Reals
8:34 wait, I just thought of something. P(N) also has countably infinite amount of elements. Why does it say here that it's the second type, the uncountable one? Or is there something I'm missing.
Why it's countable is easily proven. There exists a function that, for any element of P(N), it makes a sum of powers of 2 such that the exponents of those powers are the elements of that element. It is verifiable that that function is a bijection, therefore |P(N)| is a countable infinity as well
Why do you think P(N) has countably infinite elements? This directly contradicts Cantor's theorem.
@@EpicMathTime I put it in my comment but basically we can count them. Ø, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3} etc etc.
I thought about what I said and maybe this counting doesn't work since we cannot count all the elements with infinite elements. Maybe that is a mistake in my hypothesis?
@@fountainovaphilosopher8112 Yeah, that's a big issue. I'm not sure, but I do believe the set of /finite/ subsets of natural numbers is countable.
Consider that each real number can be represented as a decimal, which can be identified with a sequence of natural numbers.
@@EpicMathTime yes, the number of finite sets is certainly countable.
Ok, thank you. Now I see better and agree that it's uncountable. I can just apply the similar logic that is applied to prove real numbers are uncountable.
@@EpicMathTime It follows from the fact that a countable union of countable sets is countable
Very nice. Though isn't the cardinality of the power set of the aleph fixed point just itself? I could be wrong about that...
No, Cantor's Theorem still holds.
@@tomkerruish2982 good to know. Thanks
@@Quasarbooster You're welcome. What gets tricky is when you consider "the set of all sets", which seems innocent enough, but isn't. Short answer, this leads to things like Russell's Paradox and proper classes.
@@tomkerruish2982 so I've heard. The same applies to the class of all ordinals as well
@@Quasarbooster I've come across a book on category theory which went one step beyond proper classes to "illegitimate conglomerates". In fact, I think there's a set theory which goes omega levels beyond proper classes.
New video yay!!!
Interesting that the term 'real' numbers is used, as in this quantum, lumpy, discontinuous universe, in general, 'real' numbers can never exist
Well, they still exist in other ways, just like how "imaginary" numbers also exist about just as much. These are just names, and ultimately, they don't mean anything. The names that they have in many other languages probably don't even translate to this. It's more important to be able to understand how the respective sets the names refer to are defined and what properties they have.
I was advised to put a human being in the channel icon, so here we go.
Basically, could you say x^(InfinityN) =Infinity(N+1) if x>1 by a finite amount? Cause when you take then cordinality of the real, it's kinda like 10^(infinite1) and the power set of the natural is kinda like 2^(infinte1)... ?
A fun fact that I think is just beautiful:
Set of all continuous real functions has the same cardinality as reals.
True! The set of all real functions has larger cardinality, but with the restriction of continuity, we have the cardinality of the reals.
A helpful fact to show that is that, since the rationals are dense in a reals, a continuous real function is completely determined by its value at the rational numbers.
Alright bro. Diving into the topics I could never cover. Good video, just subbed :)
Thanks bro! The math of fitness collab one day.
@@EpicMathTime That'd be interesting, I'm sure you could teach me a thing or two. Let's do it!
I'm only sad you did not mention the continuum hypothesis, and took it as granted.
I didn't take the continuum hypothesis as granted. The (extended) continuum hypothesis just puts the aleph numbers and the beth numbers in agreement, but the continuum hypothesis isn't needed for the beth numbers to just... exist.
@@EpicMathTime You are right. Anyway, I think you could have mentioned it so profanes don't think the beths and the alphas are necessarily the same. That could make a good extra video.
I think you're right, a video on the continuum hypothesis would be awesome. I do regret adopting this non-standard "infinity-1" convention back when I made this video (it seemed like a good idea at the time). But yeah, these cardinals "infinity-1" correspond with the beth numbers, not the aleph numbers (without CH), and my nonstandard notation really obscured this detail.
Also, this same topic was discussed in replies to the pinned comment if you want to check that out.
The only thing i hear when you say "nope, same cardinality" is "nope, just Chuck Testa"
Hey good effort keep up the work! I have subscribed your channel ..looking forward to your new videos
Such an underrated Chanel. Please keep up the good work ❤️
Thank you! Don't worry, there's a lot more to come. ;)
It would be interesting to tell us more about those axioms required for aleph 4, 5... (or infinity 4, 5...).
I am fascinated by axioms in math because, we rely on them without any way of proving or disproving them.
Now onto even bigger cardinalities - inaccesible cardinals
Amazing! Thank you so much this made my night
So is there an infinity between Aleph 0 and Aleph 1 ?
Great question. Whether or not such an infinity exists is independent from the axioms of set theory. This is called the Continuum Hypothesis.
@@EpicMathTime it is a theorem of ZF that there is no set of cardinality "between" aleph 0 and aleph 1. First, it is well-known that a) every well-orderable set has an aleph number by definition (as the least order type), and that b) there is no aleph number between aleph 0 and aleph 1 by well-ordering of ordinals. Let A be a set. If |A| is less than or equal to aleph 1, then there is an injection between A and a set of cardinality aleph 1, meaning A is well-orderable, and thus |A| (being an aleph number) cannot be between aleph 0 and aleph 1. (Edit 2:) The messed-up thing about ZF without Choice is that there may be sets whose cardinalities aren't even comparable (this was proven to be equivalent to AC).
Edit: It is also a theorem of ZF (immediately apparent from the von Neumann ordinal definition) that aleph 1 is the cardinality of the set of all ways to well-order the naturals. The Continuum Hypothesis states that this set of well-orderings on N has the same cardinality as that of P(N), or the real numbers. With the Axiom of Choice in hand, since every set is well-orderable and thus has an aleph number, we can say succinctly that the Continuum Hypothesis states there is no cardinality between N and R, i.e., R is of cardinality aleph 1.
Maths + Cute guy = ^^^
I found it enlightening when a professor pointed out that how we think about numbers affects the cardinality of sets of numbers.
If 3 is just an object "3" then a set {3} has cardinality one. If 3 is a natural number as defined by ZF-set theory then the cardinality of {3} is 3, since 3 := {{{}}} u {{}} u {}
If 3 is taken as a real number it's usually defined as the limit of cauchy sequences of rationals (the other definitions are equivalent) and the set of all those sequences are bijective to 2^ *Q* so the cardinality of {3} is the same as the cardinality of reals so card{3} = card{ *R* }