Axiom Of Choice

Statement:

$UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)$ : The union of a denumerable family of denumerable subsets of ${\Bbb R}$ is denumerable.

Howard_Rubin_Number: 6

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:

Book references
Zermelo's Axiom of Choice, Moore, G.H., 1982

Note connections:
Note 3

The following forms are listed as conclusions of this form class in rfb1: 5, 76, 210, 324, 304, 6, 18, 34, 31, 46-K, 96, 98, 103, 124, 127, 154, 158, 243, 190, 173, 236, 198, 217, 235, 273, 237, 240, 241, 249, 285, 293, 291, 322, 323, 330, 350, 358, 357, 382, 244, 119, 238, 294, 314, 183-alpha, 59-le, 136-k, 220-p, 288-n, 342-n, 308-p, 373-n,

Howard-Rubin Number	Statement	References
6 A	Every uncountable subset of ${\Bbb R}$ contains a condensation point. \ac{G.~Moore} \cite{1982} p 205 and 324, \ac{Sierpi\'nski} \cite{1918}.
6 B	Every uncountable subset of ${\Bbb R}$ has two condensation points. \ac{G.~Moore} \cite{1982} p 205 and 324, \ac{Sierpi\'nski} \cite{1918}.
6 C	If $A\subseteq{\Bbb R}^n$ and $A\bigcap B$ is countable for every bounded $B$ then $A$ is countable. \ac{G.~Moore} \cite{1982} p 36, \ac{Keremedis/Howard/Rubin/Stanley/Tachtsis} \cite{1999}.