[FOM] The boundary of objective mathematics

[Monroe Eskew]

Your position seems inconsistent.  You say that infinite objects do
not exist, yet you permit statements whose truth value is determined
by infinite objects.  There is no finitistic proof of the consistency
of ZFC.  But you say it is an objective statement because it is
determined by checking a recursive statement for each natural number.
(i.e. Check for each n whether n codes a proof of 0=1 from ZFC.)  The
set of theorems of ZFC is recursively enumerable (r.e.) but not
recursive, hence is an infinite set for which the membership relation
cannot be reduced to a finitistic property.

You should also note that any set of n-tuples of natural numbers is
r.e. if and only if it is \Sigma_1.  Thus, many collections of natural
number n-tuples definable by a number theoretic formula with
alternating quantifiers are not r.e.  If you want to say that
quantification over natural numbers is always meaningful, then this
goes far beyond r.e. sets.

Monroe

On Mon, Mar 9, 2009 at 5:50 PM, Paul Budnik <paul at mtnmath.com> wrote:
> 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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