[FOM] The boundary of objective mathematics

[Monroe Eskew]

On Mon, Mar 9, 2009 at 5:50 PM, Paul Budnik <paul at mtnmath.com> wrote:
> Real numbers are problematic because they seem so natural and obvious. I
> think we can talk about an arbitrary integer parameter for a recursive
> process. And I think we can talk about an arbitrary infinite sequence of
> integer parameters. This all makes sense to me in an always finite but
> potentially infinite universe. Asking what a TM will do for all possible
> infinite sequences of inputs make sense because every event that
> determines that can be recursively enumerated. However I think this
> question is a human creation. It is not a question about something that
> can ever exist. It only asks about the unbounded future of a recursive
> process in a potentially infinite universe.
>
> In contrast the CH asks about what infinite sets exist in an absolute
> sense. The question is meaningful relative to the sets provably
> definable in a formal system but it has no absolute meaning because
> infinite sets are human creations and abstractions.

This depends on your view of the physical universe.  If space is
continuous, then CH can be viewed as a statement quantifying over
regions of space.  Granted it will not be experimentally verifiable
since our instruments and sense organs do not have the required
precision.  But you can view CH as saying the following:

Let AB be a line segment.  Construct a square on AB, call it ABCD.
For every subregion of AB, there is a subregion X of the square ABCD
such every line perpedicular to AB intersects X at exactly one point,
and the projection of X onto AC is either the entire segment AC, or
the set of rational division points of AC.



> I only demand a
> constructive proof that all the events that determine the statement are
> themselves determined by finite events.
>

Why not iterate this, allowing allowing statements to be determined by
statements determined by finite events, statements determined by
statements determined by statements determined by finite events, and
so on?

Monroe