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. 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.
