Cardinality bigger than that of the reals

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SUMMARY

The discussion centers on the concept of constructing a beth-2 set, which represents the cardinality of the set of all functions from the reals to the reals. Participants explore the implications of cardinality, emphasizing that the power set of any set has a higher cardinality than the original set. The iterative process of taking power sets leads to increasingly larger cardinalities, establishing a clear hierarchy in set theory.

PREREQUISITES
  • Understanding of cardinality in set theory
  • Familiarity with power sets and their properties
  • Knowledge of functions and mappings in mathematics
  • Basic concepts of infinite sets and their classifications
NEXT STEPS
  • Research the properties of beth numbers in set theory
  • Explore the implications of Cantor's theorem on cardinality
  • Study the construction and characteristics of power sets
  • Investigate the concept of functions from reals to reals in advanced mathematics
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Mathematicians, students of set theory, and anyone interested in the foundations of mathematics and cardinality concepts.

JamesOrland
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I was wondering. Has anyone ever been able to build a beth-2 set? What does it look like? What could it possibly look like?
 
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The set of all mappings (functions) from the reals to the reals.
 
JamesOrland said:
I was wondering. Has anyone ever been able to build a beth-2 set? What does it look like? What could it possibly look like?

The power set is always of higher cardinality than the original set. Then you can take the power set of the power set, etc.
 

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