> On 16 Dec 2018, at 22:28, 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] >> <mailto:[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] >>> <mailto:[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:
Then we abandon it. Here we talk about arithmetical theories. Everyone believe RA is consistent. Every mathematician but one (Nelson) believe PA is consistent. > > axiom 1: Trump is a genius. > axiom 2: Trump is stable. That is not an axiomatic theory. Bruno > > theorem: Trump is a stable genius. > > Brent > > -- > 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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.
