Cardinality of Sets: N & Omega Explained

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Discussion Overview

The discussion revolves around the concept of cardinality, specifically focusing on the set denoted as [N]^{\omega} and the meaning of omega in this context. Participants explore the relationship between ordinals and cardinals, and the implications of these concepts in set theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of omega in the context of the cardinality of [N]^{\omega}.
  • Another participant suggests that omega might relate to the function space X^Y, indicating a potential connection to set functions.
  • It is noted that lowercase omega is typically understood as the first infinite ordinal.
  • There is a query regarding whether omega is equivalent to aleph naught, with a suggestion that aleph naught raised to the power of aleph naught results in aleph naught.
  • A clarification is made that while omega's cardinality is aleph naught, omega itself is an ordinal, contrasting with aleph naught as a cardinal.
  • One participant describes [N]^{\omega} as an infinite sequence of natural numbers.
  • Another participant defines [N]^{\omega} as the collection of subsets of N that have a size of omega.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between omega and aleph naught, with some asserting they are equivalent while others clarify their distinct roles as ordinal and cardinal. The discussion remains unresolved regarding the implications of these concepts.

Contextual Notes

There are limitations in the discussion regarding the definitions and properties of ordinals and cardinals, as well as the mathematical steps involved in the relationships being explored.

saadsarfraz
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The cardinality of set of [N][tex]\omega[/tex] . what does omega stands for?
 
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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] ?
 
The lower case omega is (usually) the first infinite ordinal.
 
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?
 
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.
 
It's an infinite sequence of natural numbers.
 
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]
 

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