[FOM] The boundary of objective mathematics
Paul Budnik
paul at mtnmath.com
Mon Mar 9 20:50:30 EDT 2009
Monroe Eskew wrote:
> If you believe the power set operation is a determinate concept,
> despite its indescribability in first order logic, the the continuum
> hypothesis would have an objective truth value. ...
I do not "believe the power set operation is a determinate concept". I
do not think infinite objects exists although I think it possible that
our universe may continue forever and be potentially infinite. This is a
very old approach to mathematical infinity.
> Set theorists who
> take a "combinatorial" view of sets would fall into this camp. Also,
> anyone who believes that the set of natural numbers is determinate
> would agree that any number-theoretic statement has an objective truth
> value, even those that are not provable from the incomplete theory PA.
> There are also number-theoretic statements that are not decided by
> ZFC or any consistent recursive extension. But the theory of natural
> numbers is generically absolute, in that it is not affected by
> forcing, unlike CH. This lends it some claim of objectivity, though
> perhaps a weaker one, since generic absoluteness is weaker than
> recursive decidability.
>
I am not suggesting that properties have to be recursively decidable,
only that they be determined by events which are recursively enumerable.
The halting problem is not recursively decidable but is determined by a
recursively enumerable set of events (what the TM does at each time step).
I think most of commonly used mathematics and thus most, if not all, of
number theory will meet the condition that a statement be determined by
a recursively enumerable set of events. This goes for number theoretic
statements not provable in ZFC including most obviously the consistency
of ZFC. It is straight forward to enumerate the events that determine this.
Paul Budnik
www.mtnmath.com
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