[FOM] Completeness formulations
Brian Hart
hart.bri at gmail.com
Mon Oct 13 13:45:53 EDT 2008
Barrow wrote an interesting paper in which he claimed that physical
incompleteness would limit physicists in terms of their ability to
make predictions within a certain physio-theoretic context, but that
no inhibition would be felt by the Theorists of Everything as a
result.
On Sat, Oct 11, 2008 at 6:59 PM, Harvey Friedman
<friedman at math.ohio-state.edu> wrote:
>
> On Oct 9, 2008, at 10:43 PM, Brian Hart wrote:
>
>>
>> Why doesn't Godel's 1st Incompleteness Theorem imply the
>> incompleteness of any theory of physics T, assuming that T is
>> consistent and uses arithmetic? Shouldn't the constructors of the
>> Theory of Everything be alarmed? I know this suggestion of
>> application of Godel's theorem was made decades ago but why didn't it
>> make a bigger impact? Is it because it is wrong or were there some
>> sociological reasons for mainstream ignorance of it?
>> _
>
> This question suggests the following formulation.
>
> To focus matters, let's assume that we are talking about a theory
> where the mathematics involved is put in purely set theoretic terms.
> Of course, there are modifications of the setup below that may be
> preferable for some purposes.
>
> So we are looking at first order theories T in many sorted predicate
> calculus with a sort for sets, with epsilon between sets, and equality
> between sets.
>
> We have the following concepts.
>
> COMPLETENESS_1. Any two model of T in which the sort for sets is
> interpreted as "all sets, with ordinary membership and equality"
> satisfy the same sentences.
>
> COMPLETENESS_2. Any two models of T in which the sort for sets is
> interpreted as "all sets, with ordinary membership and equality" are
> isomorphic.
>
> We can also allow second and higher order theories T, with the usual
> second and higher order semantics.
>
> Now these formulations are obviously made in class theory. If we want
> formulations in set theory, then we would not use the entire set
> theoretic universe for the set theoretic sort, but instead some
> fragment thereof, renaming the sort as the such and such restricted
> set sort. E.g., the set sort might be "the hereditarily finite set
> sort", or "the rank < omega + omega set sort".
>
> Here is a semi formal conjecture:
>
> CONJECTURE. For any reasonably system that purports to formalize some
> physical theory incorporating mathematics, if it is Complete_1 then it
> is Complete_2. Furthermore, if it is not Complete_1, then there is a
> simple sentence that violates Complete_1.
>
> In particular, this Conjecture suggests that if we look at formalized
> physical theories, and we factor out the math incompleteness, then we
> get completeness issues that are of a totally different character than
> those in mathematics.
>
> Harvey Friedman
>
>
>
>
>
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