Axiom Of Choice

Statement:

\(\aleph_{1}\) is regular.

Howard_Rubin_Number: 34

Parameter(s): This form does not depend on parameters

This form's transferability is: Transferable

This form's negation transferability is: Negation Transferable

Article Citations:
Russell-1906: On some difficulties in the theory of transfinite numbers and order types

Book references

Note connections:
Note 3

The following forms are listed as conclusions of this form class in rfb1: 5, 19, 38, 104, 419,

Howard-Rubin Number	Statement	References
34 A	\(\exists\alpha < \omega_2\) such that there are no hereditarily countable sets of rank \(\alpha\).	Jech [1982] Note [48]
34 B	\(\omega_1\) with the order topology is not paracompact.	Good-Tree-1995 Note [107]
34 C	\(\omega_1\) with the order topology has the property that every infinite subset has a limit point in \(\omega_1\).	Good-Tree-1995 Note [107]
34 D	\(\omega_1\) with the order topology does not contain a countable discrete family of open subsets.	Good-Tree-1995 Note [107]
34 E	\(\omega_1\) with the order topology is not weakly Lindelöf.	Good-Tree-1995 Note [43] Note [107]