英和和英

使用する際の注意点

Some related concepts:

• The aleph numbers, ${\displaystyle \aleph _{0},\aleph _{1},\dots }$, represent an enumeration of the transfinite numbers.
  • The smallest transfinite number, ${\displaystyle \aleph _{0}}$ (aleph-null) — also denoted ${\displaystyle \omega }$ — is the cardinality of the natural numbers. Each succeeding ${\displaystyle \aleph _{n}}$ is defined to be the smallest transfinite number greater than ${\displaystyle \aleph _{n-1}}$.
• The beth numbers, ${\displaystyle \beth _{0},\beth _{1}\dots }$, are an enumerated subset of the transfinite numbers, defined in a different, in some ways more mathematically tractable way. It is hypothesised that the beth numbers are in fact precisely the aleph numbers.
  • By definition, ${\displaystyle \beth _{0}=\aleph _{0}}$ and, for ${\displaystyle n>0}$, ${\displaystyle \beth _{n}}$ is the power set of ${\displaystyle \beth _{n-1}}$. A consequence of this definition is that ${\displaystyle \beth _{1}}$ is the cardinality of the real numbers.
• It is not immediately clear that an ordered enumeration of the transfinite numbers, such as the aleph numbers represent, is even possible. In particular, it depends upon the axiom of choice (historically controversial for infinite collections of sets), without which transfinite numbers greater than ${\displaystyle \aleph _{0}}$ might exist that are not mutually comparable.
• The continuum hypothesis states that there is no transfinite number between the cardinality of the natural numbers and that of the real numbers — i.e., that the cardinality of the real numbers is ${\displaystyle \aleph _{1}}$.
  • The continuum hypothesis implies that ${\displaystyle \beth _{1}=\aleph _{1}}$.
  • The generalized continuum hypothesis states that ${\displaystyle \beth _{n}=\aleph _{n}}$.

下位語

• aleph number, beth number
  • aleph-null, aleph-one
  • ω