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. 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.
