> On 16 Dec 2018, at 22:28, Brent Meeker <meeke...@verizon.net> wrote: > > > > On 12/15/2018 10:24 PM, Jason Resch wrote: >> >> >> On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker <meeke...@verizon.net >> <mailto:meeke...@verizon.net>> wrote: >> >> >> On 12/15/2018 6:07 PM, Jason Resch wrote: >>> >>> >>> On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker <meeke...@verizon.net >>> <mailto:meeke...@verizon.net>> 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
