On Mon, Dec 17, 2018 at 12:05 AM Brent Meeker <[email protected]> wrote:
> > > On 12/16/2018 9:36 PM, Jason Resch wrote: > > > > On Sun, Dec 16, 2018 at 10:22 PM Brent Meeker <[email protected]> > wrote: > >> >> >> On 12/16/2018 4:39 PM, Jason Resch wrote: >> >> >> >> On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker <[email protected]> >> wrote: >> >>> >>> >>> On 12/16/2018 1:56 PM, Jason Resch wrote: >>> >>> >>> >>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <[email protected]> >>> wrote: >>> >>>> >>>> >>>> On 12/15/2018 10:24 PM, Jason Resch wrote: >>>> >>>> >>>> >>>> On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker <[email protected]> >>>> wrote: >>>> >>>>> >>>>> >>>>> On 12/15/2018 6:07 PM, Jason Resch wrote: >>>>> >>>>> >>>>> >>>>> On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker <[email protected]> >>>>> wrote: >>>>> >>>>>> >>>>>> >>>>>> On 12/15/2018 5:42 PM, Jason Resch wrote: >>>>>> >>>>>> hh, but diophantine equations only need integers, addition, and >>>>>>> multiplication, and can define any computable function. Therefore the >>>>>>> question of whether or not some diophantine equation has a solution can >>>>>>> be >>>>>>> made equivalent to the question of whether some Turing machine halts. >>>>>>> So >>>>>>> you face this problem of getting at all the truth once you can define >>>>>>> integers, addition and multiplication. >>>>>>> >>>>>>> >>>>>>> There's no surprise that you can't get at all true statements about >>>>>>> a system that is defined to be infinite. >>>>>>> >>>>>> >>>>>> But you can always prove more true statements with a better system of >>>>>> axioms. So clearly the axioms are not the driving force behind truth. >>>>>> >>>>>> >>>>>> And you can prove more false statements with a "better" system of >>>>>> axioms...which was my original point. So axioms are not a "force behind >>>>>> truth"; they are a force behind what is provable. >>>>>> >>>>>> >>>>> There are objectively better systems which prove nothing false, but >>>>> allow you to prove more things than weaker systems of axioms. >>>>> >>>>> >>>>> By that criterion an inconsistent system is the objectively best of >>>>> all. >>>>> >>>>> >>>> The problem with an inconsistent system is that it does prove things >>>> that are false i.e. "not true". >>>> >>>> >>>>> However we can never prove that the system doesn't prove anything >>>>> false (within the theory itself). >>>>> >>>>> >>>>> You're confusing mathematically consistency with not proving something >>>>> false. >>>>> >>>> >>>> They're related. A system that is inconsistent can prove a statement >>>> as well as its converse. Therefore it is proving things that are false. >>>> >>>> >>>> But a system that is consistent can also prove a statement that is >>>> false: >>>> >>>> axiom 1: Trump is a genius. >>>> axiom 2: Trump is stable. >>>> >>>> theorem: Trump is a stable genius. >>>> >>> >>> So how is this different from flawed physical theories? >>> >>> >>> The difference is that mathematicians can't test their theories. >>> >> >> Sure they can: A set of axioms predicts a Diophantine equation has no >> solutions. We happen to find it does have a solution. We can reject that >> set of axioms. >> >> >> Then the axioms must have also included enough to include Diophantine >> equations (e.g. PA) so you have added axioms making the system inconsistent >> and every proposition is a theorem. The only test of the theory was that >> it is inconsistent. >> > > There is also soundness <https://en.wikipedia.org/wiki/Soundness> which I > think more accurately reflects my example above. > > > "...a system is sound when all of its theorems are tautologies." Which is > to say it is true that the theorem follows from the axioms. Not that it is > true simpliciter. > How about this: Arithmetic soundness[edit <https://en.wikipedia.org/w/index.php?title=Soundness&action=edit§ion=5> ] If *T* is a theory whose objects of discourse can be interpreted as natural numbers <https://en.wikipedia.org/wiki/Natural_numbers>, we say *T* is *arithmetically sound* if all theorems of *T* are actually true about the standard mathematical integers. For further information, see ω-consistent theory <https://en.wikipedia.org/wiki/%CE%A9-consistent_theory>. Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
