Cardinality of Sets: N & Omega Explained

In summary, the cardinality of the set of [N]^{\omega} is aleph naught, which is the first infinite ordinal. Omega is often used to represent this concept, but it is not the same as aleph naught. The notation [N]^{\omega} typically denotes the collection of subsets of N with a size of omega.
  • #1
saadsarfraz
86
1
The cardinality of set of [N][tex]\omega[/tex] . what does omega stands for?
 
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  • #2
This seems a bit out of the blue, I never seen something like that but could it by any chance be referring to [itex]X^Y=\{f : Y \rightarrow X \}[/itex] ?
 
  • #3
The lower case omega is (usually) the first infinite ordinal.
 
  • #4
so is omega the same as aleph not. and in this case it would be aleph not to the power aleph not which gives aleph not?
 
  • #5
saadsarfraz said:
so is omega the same as aleph not. and in this case it would be aleph not to the power aleph not which gives aleph not?

omega's cardinality is aleph naught. But omega is an ordinal, while aleph naught is a cardinal.
 
  • #6
It's an infinite sequence of natural numbers.
 
  • #7
Usually the notation [itex][N]^{\omega}[/itex] denotes the collection of subsets of N of size [itex]\omega[/itex], i.e.:

[tex][N]^{\omega} = \{ X \subseteq N : |X| = \omega \}[/tex]
 

1. What is the cardinality of a set?

The cardinality of a set is the number of elements or objects in that set. In other words, it represents the size or quantity of the set.

2. What is the difference between finite and infinite sets?

A finite set is a set that has a specific and definite number of elements. An infinite set, on the other hand, has an unlimited or endless number of elements.

3. What is the cardinality of the set of natural numbers (N)?

The cardinality of the set of natural numbers (N) is infinite, as there is no largest natural number and the set continues infinitely in both directions.

4. What is the cardinality of the set of counting numbers (N0)?

The cardinality of the set of counting numbers (N0) is also infinite, as it includes all the natural numbers along with zero.

5. What is the difference between the cardinalities of N and Ω?

The cardinality of N is equal to the cardinality of Ω (the set of all natural numbers), as both sets have an infinite number of elements. However, the cardinality of Ω is greater than the cardinality of N as it also includes non-natural numbers such as fractions, decimals, and irrational numbers.

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