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.
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.
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