On 2/20/2014 5:39 AM, email@example.com wrote: > > > Zeitgeist on the odds of my students to understand transfinite set theory:" if they can be convinced that what their senses tell them may be Not be the whole picture, then they may have a chance." > > Virgil, appearing as Wisely Non-Theist: "There are 'more' real numbers than there are finite definitions to define them with, so most reals can only be defined collectively, not individually." > > Ben Bacarrisse emphasized: "They are not 'entire undefinable'. The set of them can be defined." And he added: "You can know things about the set. For example, that it can't be bijected with N." > > William P. Hughes: "A subcollection of a listable collection may not be listable." > > Alan Smaill: "After all, since matheology accepts undefinable real numbers, then why are you trying to suggest that it does not accept undefinable enumerations?" > > That is true. I never got a grasp of this idea: Why should undefinable definable enumerations (aka lists) be exemptet from the list of unlisted exemptions? > > Regards, WM >
In my other response, I tried to reduce your question to the epistemological concerns of predicative mathematics.
On that account, the objects that comprise a function must exist prior to the definite determination of the function itself. So, either you have incomplete definitions or incomplete systems in which the notion of definition cannot exhause the possibilities and there is something that cannot be defined.
If one interprets your attempts to place "all numbers" into the list of finite numbers as a tacit form of recognizing a system of incomplete definitions in favor of an incomplete system, then your objections have some merit. The problem is that you want to have it both ways. That is not possible.