🚀 go-pugleaf

In article <1992Nov16.131202.17710@sei.cmu.edu>, firth@sei.cmu.edu (Robert Firth) writes:
> In article <350@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
> 
>...
> >theorems. The names of all real numbers created by such a program are
> >obviously countable.
> 
> Well, yes, the names of any set of things created by an "enumeration"
> program muct be countable, by definition.  But that's not what you
> originally claimed.  You said:
> 
> >For example the real numbers definable in any consistent formal system
> >are countable.
> 
> By what sleight of hand did "definable" change into "enumerable"?  Isn't
> that the old constructivist premise (or fallacy, as most of us think).

A real number is not definable in a formal system unless it is named
in some theorem in that system. The reals definable in a formal system
are limited to these. Of course there are always additional reals not
definable in the system that still satisfy the definition of a real number
in the system. Mathematicians who believe there is a Platonic heaven
of completed infinite totalities intend for their definition of reals to
encompass all residents of this heaven that satisfy that definition.
This is a philosophical not a mathematical definition and a highly
questionable one in my opinion.

The constructivist position is different. It does not accept proof
by contradiction as a valid argument for the existence of a mathematical
object. The constructivist position involves more than this and has
variations, but this is a universal requirement. Reals definable in
this way are nameable and thus definable in the sense
I used the term.

Follow ups are directed to sci.logic.

Paul Budnik