[FOM] The boundary of objective mathematics
Monroe Eskew
meskew at math.uci.edu
Sat Mar 7 11:32:58 EST 2009
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. 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.
On Fri, Mar 6, 2009 at 10:28 AM, Paul Budnik <paul at mtnmath.com> wrote:
>
> I have long felt that objective mathematics is limited to statements
> determined by a recursively enumerable sequence of events, because such
> statements can, at least in theory, be determined by events that occur
> physically.
>
> Some statements requiring quantification over the reals meet this
> definition. For example we can ask if a TM that keeps requesting integer
> inputs will eventually halt regardless of what sequence of integer
> inputs it receives. The result is determined by what the TM does for
> every possible finite sequence of integer inputs. Such questions may be
> of practical interest for those existing in an always finite, but
> potentially infinite universe. (It is possible that we inhabit such a
> universe in spite of what cosmology predicts for our ultimate fate. Of
> necessity cosmology is an extremely speculative science.) Thus one might
> be interested to know if the human species will evolve an infinite chain
> of descendant species. One might even wish to make decisions that make
> this more likely.
>
> The Continuum Hypothesis is an obvious example of a statement that does
> not meet this condition. Intuitionists are more conservative (at least
> in terms of what proofs they accept) but most mathematicians are, I
> suspect, far more liberal in the mathematics they believe is objective.
> Yet there is increasing scepticism about the objective truth of the
> Continuum Hypothesis and similar statements.
>
> For those that question the objectivity of the Continuum Hypothesis,
> what do you think of this proposal for objective mathematics. If the
> answer is not much, where would you draw the boundary and why?
>
> Paul Budnik
> www.mtnmath.com
>
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