Defining Aleph-One: The Elusive Cardinal Number

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Homework Help Overview

The discussion revolves around the definition of aleph-one, particularly in the context of set theory and cardinal numbers. The original poster expresses confusion regarding its definition and relationship to aleph-zero and the continuum hypothesis.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of aleph-one and its relationship to other cardinal numbers, questioning the reliance on specific set theories. There is mention of varying interpretations based on acceptance of the continuum hypothesis.

Discussion Status

The conversation is ongoing, with participants sharing insights about the definitions and implications of aleph-one. Some guidance has been offered regarding the relationship between aleph-one and the continuum hypothesis, but no consensus has been reached.

Contextual Notes

There is a noted lack of clarity in the original poster's reference materials, and assumptions about set theory frameworks are being questioned. The discussion reflects uncertainty about the definitions and implications of different cardinalities.

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Homework Statement


aleph-0 is defined as the cardinality as the integers right?

How is aleph-one defined? Is the only definition as the next largest cardinal number after aleph-1?

Annoyingly, my set theory book (kaplansky) doesn't seem to define it anywhere. Also the wikipedia article just confused me more.


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The Attempt at a Solution

 
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How much do you know about ordinal numbers?
 


I see now. The problem was that Kaplansky uses alpha-two on page 47 but defines ordinal numbers on page 57.
 


Surely one of the real problems with defining aleph-1 is that what it is depends on what set theory you use. If you accept CH, then it is the cardinality of the real numbers. But it need not be. I have this idea that there may be infinitely many cardinals between aleph-0 and c (cardinality of the continuum, or 2^aleph-0) in some models, but it is just a feeling.
 

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