Aleph Omega = Aleph infinity?

[CNBA3]

I was wondering about Aleph numbers, particularly Aleph Omega which is iiu basically every uncountable Aleph number till infinity, would it be another word for Aleph infinity? I was readying somewhere here about someone saying that the two are synonymous in that sense.

Alephs are basically infinities of larger size/sets, so I would like people who are qualified to clarify it please? Thank you.

source

https://vsbattles.com/threads/aleph-omega.126028/

 

[Cat275]

Bump or no?

 

[CNBA3]

Bump or no?

I like to have it at a bump just to hope for any attention

 

[Cat275]

I like to have it at a bump just to hope for any attention

So i take it you are still looking for answers?

 

[CNBA3]

So i take it you are still looking for answers?

Hopefully.

 

[XXKINGXX69]

Why don't you ask a math exprt

 

[CNBA3]

Why don't you ask a math exprt

I did, sent a private message to someone who talked about this in the past on vsbattle, no reply yet

 

[Cat275]

Anyways what is the question here again?

All i can say right now is that ℵW (i can't type the symbol) Is aleph infinite because
W is a infinite denotation of ordinals.
(0,1,2,3,,,,,,,,,,,,,,,,,,,,,,,9278 and so on.)

So it should mean ℵW Would be a infinite ordinal aleph.

I could give more details later if you want i guess but this is a simple explanation of it....
(+ im lazy)

Idk if there is something called aleph infinite though but aleph omega should be akin to it.

(Yeah i got lazy so i didn't manage to answer this the whole time but now i got enough energy to answer this.... time to be lazy again though.)

 

[CNBA3]

Anyways what is the question here again?

All i can say right now is that ℵW (i can't type the symbol) Is aleph infinite because
W is a infinite denotation of ordinals.
(0,1,2,3,,,,,,,,,,,,,,,,,,,,,,,9278 and so on.)

So it should mean ℵW Would be a infinite ordinal aleph.

I could give more details later if you want i guess but this is a simple explanation of it....
(+ im lazy)

Idk if there is something called aleph infinite though but aleph omega should be akin to it.

(Yeah i got lazy so i didn't manage to answer this the whole time but now i got enough energy to answer this.... time to be lazy again though.)

Thanks, I just wanted to understand if the two are akin to each other if say somewhere someone like “aleph infinity” or akin to that in context as well that it would be viable used.

 

[Kowalski]

I was wondering about Aleph numbers, particularly Aleph Omega which is iiu basically every uncountable Aleph number till infinity, would it be another word for Aleph infinity? I was readying somewhere here about someone saying that the two are synonymous in that sense.

Alephs are basically infinities of larger size/sets, so I would like people who are qualified to clarify it please? Thank you.

source

https://vsbattles.com/threads/aleph-omega.126028/

Aleph-infinity is probably just a former name for aleph-omega.

 

[Cat275]

Thanks, I just wanted to understand if the two are akin to each other if say somewhere someone like “aleph infinity” or akin to that in context as well that it would be viable used.

They should be akin to another maybe even the same considering the fact that an omega well orderly denotates the whole natural number line.

(maybe real number line as well, i don't think so though.)

So aleph omega is bassically the same denotation as omega except it includes aleph so a well ordered infinite aleph sets.

(I guess it's an infinite aleph ordinal or it's bassically just a aleph version of omega as the name already implies it.)

So in short (and as a laymens term)
Aleph omega is bassically the infinite denotation and order of a infinite set.
(Specifically the aleph set.)

Hope you get the gist of it.

 

[CNBA3]

Aleph-infinity is probably just a former name for aleph-omega.

They should be akin to another maybe even the same considering the fact that an omega well orderly denotates the whole natural number line.

(maybe real number line as well, i don't think so though.)

So aleph omega is bassically the same denotation as omega except it includes aleph so a well ordered infinite aleph sets.

(I guess it's an infinite aleph ordinal or it's bassically just a aleph version of omega as the name already implies it.)

So in short (and as a laymens term)
Aleph omega is bassically the infinite denotation and order of a infinite set.
(Specifically the aleph set.)

Hope you get the gist of it.

okay, thank you both!

 

[Achille]

I was wondering about Aleph numbers, particularly Aleph Omega which is iiu basically every uncountable Aleph number till infinity, would it be another word for Aleph infinity? I was readying somewhere here about someone saying that the two are synonymous in that sense.

Alephs are basically infinities of larger size/sets, so I would like people who are qualified to clarify it please? Thank you.

source

https://vsbattles.com/threads/aleph-omega.126028/

Aleph omega, denoted by אω, is a cardinal number in set theory, a branch of mathematics that studies the properties of sets and their relationships. Aleph omega is the first uncountable cardinal number and is often used as a standard for comparison of cardinalities of sets. It is the smallest cardinality of a set that cannot be put in one-to-one correspondence with the set of natural numbers (which has cardinality א₀).

Aleph omega can be described as the cardinality of the set of all countable ordinal numbers. An ordinal number is a well-ordered set that is similar to the natural numbers, but can be used to describe the order type of other sets as well. The set of all countable ordinal numbers is called the class of countable ordinals and is denoted by On.

Aleph omega is the first uncountable cardinal, meaning that it is strictly larger than the cardinality of any countable set. This is because it is the cardinality of the set of all countable ordinal numbers, which are uncountable in the sense that they cannot be put into a one-to-one correspondence with the natural numbers.

One important property of aleph omega is that it is the first cardinal for which the Axiom of Choice is equivalent to its existence. The Axiom of Choice is a statement in set theory that asserts the existence of a choice function for any collection of non-empty sets. In other words, given any collection of non-empty sets, there exists a function that selects one element from each set. The existence of aleph omega is equivalent to the Axiom of Choice, meaning that either both are true or both are false.

Aleph omega also has important connections to the concept of immeasurable cardinals. An immeasurable cardinal is a cardinal number that is not the cardinality of a subset of any given set of smaller cardinality. Immeasurable cardinals are related to aleph omega because they can be used to prove the existence of aleph omega. Specifically, if there exists an immeasurable cardinal, then it is possible to construct a set of cardinality אω by using the Axiom of Choice.

The relationship between alephs, aleph omega, and immeasurable cardinals can be expressed mathematically as follows:

Let א be the class of all cardinal numbers, and let On be the class of all countable ordinal numbers. Then:

אω = cardinality of On.

If α is an immeasurable cardinal, then אω = α.

If there exists an immeasurable cardinal, then אω exists.

 

[Ben_CleverName]

Ain't no way that this thread popped up when I Google searched aleph-omega
The above text is blatantly copied from ChatGPT given its sentence structure, and it's also blatantly wrong. It literally says that aleph-omega is the first uncountable cardinal number???
The first smallest uncountable cardinal is aleph-1. The second is aleph-2. Assuming CH, you can say that aleph-2 is the powerset of aleph-1, whereas aleph-3 is the powerset of aleph-2, etc.. Aleph-omega is basically the infinity-th of that stretch of numbers.

 

[KingNanaya]

these numbers hurt

 

[GarrixianXD]

Pretty much, yeah. Infinity is basically to denote an extreme and absolute limit in calculus and algebra; not a number but rather a concept. Omega falls into the same category in set theory since it is an infinite ordinal.