SUMMARY
Aleph-one (ℵ₁) is defined as the next largest cardinal number after aleph-zero (ℵ₀), which represents the cardinality of the integers. The definition of aleph-one varies depending on the set theory framework employed. Under the Continuum Hypothesis (CH), aleph-one is equivalent to the cardinality of the real numbers. However, in some models of set theory, there may exist infinitely many cardinal numbers between aleph-zero and the cardinality of the continuum (c, or 2^ℵ₀).
PREREQUISITES
- Understanding of cardinal numbers and their properties
- Familiarity with ordinal numbers and their definitions
- Knowledge of the Continuum Hypothesis (CH)
- Basic concepts of set theory as presented in texts like Kaplansky's
NEXT STEPS
- Study the implications of the Continuum Hypothesis in set theory
- Explore the differences between cardinal and ordinal numbers
- Investigate models of set theory that allow for multiple cardinals between ℵ₀ and c
- Read Kaplansky's set theory book for a deeper understanding of ordinal definitions
USEFUL FOR
Mathematicians, students of set theory, and anyone interested in the foundations of mathematics and cardinality concepts.