In article <email@example.com>, firstname.lastname@example.org wrote:
> A real number is defined by (at least) one or (frequently) more of those > expressions. Therefore all real numbers form a subset of the set of all > finite expressions. According to ZF set theory a subset of a countable set is > countable.
That presumes that there must be at least one definition for each real number. Which maybe the case in WM's wild weird world of WMytheology but is provably not the case in standard mathematics.
The set of all finite expressions, whose existence WM cites above, does not contain enough expressions to define individually each and every subset of itself, much less also define individually each and every member of the set of all real numbers. --