On 12/16/2018 11:02 PM, Jason Resch wrote:
On Mon, Dec 17, 2018 at 12:05 AM Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:On 12/16/2018 9:36 PM, Jason Resch wrote:On Sun, Dec 16, 2018 at 10:22 PM Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote: On 12/16/2018 4:39 PM, Jason Resch wrote:On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote: On 12/16/2018 1:56 PM, Jason Resch wrote:On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <meeke...@verizon.net <mailto: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: 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."...a system is sound when all of its theorems are tautologies." Which is to say it is true that the theorem follows from the axioms. Not that it is true simpliciter. How about this: Arithmetic soundness[edit <https://en.wikipedia.org/w/index.php?title=Soundness&action=edit§ion=5>] If/T/is a theory whose objects of discourse can be interpreted asnatural numbers <https://en.wikipedia.org/wiki/Natural_numbers>, we say/T/is/arithmetically sound/if all theorems of/T/are actually true about the standard mathematical integers. For further information, seeω-consistent theory <https://en.wikipedia.org/wiki/%CE%A9-consistent_theory>.
OK. But note that it assumes the natural numbers exist independently of T and in order to be applied it assumes one can know that every theorem of T is true of the natural numbers. If the natural numbers exist independently of T then a theorem of T may or may not apply to them. But in fact the natural numbers are only defined as that which satisfies some theory T'.
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